[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifIntegrable (f + g) p μ
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hε2 : 0 < ε / 2 := half_pos hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, fun i s hs hμs => _⟩
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (indicator s ((f + g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp_rw [Pi.add_apply, Set.indicator_add']
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (indicator s (f i) + indicator s (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' (snorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans _
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (indicator s (f i)) p μ + snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by
  rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
[PROOFSTEP]
rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
⊢ snorm (indicator s (f i)) p μ + snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [hε_halves]
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ
hε : 0 < ε
hε2 : 0 < ε / 2
δ₁ : ℝ
hδ₁_pos : 0 < δ₁
hfδ₁ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
δ₂ : ℝ
hδ₂_pos : 0 < δ₂
hgδ₂ :
  ∀ (i : ι) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
⊢ snorm (indicator s (f i)) p μ + snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
[PROOFSTEP]
exact
  add_le_add (hfδ₁ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_left _ _))))
    (hgδ₂ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_right _ _))))
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
⊢ UnifIntegrable (-f) p μ
[PROOFSTEP]
simp_rw [UnifIntegrable, Pi.neg_apply, Set.indicator_neg', snorm_neg]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
⊢ ∀ ⦃ε : ℝ⦄,
    0 < ε →
      ∃ δ h,
        ∀ (i : ι) (s : Set α),
          MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact hf
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifIntegrable (f - g) p μ
[PROOFSTEP]
rw [sub_eq_add_neg]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hg : UnifIntegrable g p μ
hp : 1 ≤ p
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
⊢ UnifIntegrable (f + -g) p μ
[PROOFSTEP]
exact hf.add hg.neg hp hf_meas fun i => (hg_meas i).neg
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
⊢ UnifIntegrable g p μ
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hfδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδ_pos, fun n s hs hμs => (le_of_eq <| snorm_congr_ae _).trans (hfδ n s hs hμs)⟩
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hfδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ indicator s (g n) =ᵐ[μ] indicator s (f n)
[PROOFSTEP]
filter_upwards [hfg n] with x hx
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hfδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : f n x = g n x
⊢ indicator s (g n) x = indicator s (f n) x
[PROOFSTEP]
simp_rw [Set.indicator_apply, hx]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSpace α
p : ℝ≥0∞
f : ι → α → β
ε : ℝ
x✝² : 0 < ε
i : ι
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑0 s ≤ ENNReal.ofReal 1
⊢ snorm (indicator s (f i)) p 0 ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
⊢ Tendsto (fun M => indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
[PROOFSTEP]
refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ indicator {x | ↑n ≤ ↑‖f x‖₊} f x = 0
[PROOFSTEP]
rw [Set.indicator_of_not_mem]
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ¬x ∈ {x | ↑n ≤ ↑‖f x‖₊}
[PROOFSTEP]
simp only [not_le, Set.mem_setOf_eq]
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑‖f x‖₊ < ↑n
[PROOFSTEP]
refine' lt_of_le_of_lt (Nat.le_ceil _) _
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑⌈↑‖f x‖₊⌉₊ < ↑n
[PROOFSTEP]
refine' lt_of_lt_of_le (lt_add_one _) _
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑⌈↑‖f x‖₊⌉₊ + 1 ≤ ↑n
[PROOFSTEP]
norm_cast
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have htendsto : ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => {x | (M : ℝ) ≤ ‖f x‖₊}.indicator f x) atTop (𝓝 0) :=
  univ_mem' (id fun x => tendsto_indicator_ge f x)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({x | (M : ℝ) ≤ ‖f x‖₊}.indicator f) μ :=
  by
  intro M
  apply hf.1.indicator
  apply
    StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
⊢ ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
[PROOFSTEP]
intro M
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
M : ℕ
⊢ AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
[PROOFSTEP]
apply hf.1.indicator
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
M : ℕ
⊢ MeasurableSet {x | ↑M ≤ ↑‖f x‖₊}
[PROOFSTEP]
apply
  StronglyMeasurable.measurableSet_le stronglyMeasurable_const
    hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ :=
  by
  rw [memℒp_one_iff_integrable] at hf
  exact hf.norm.2
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ HasFiniteIntegral fun x => ‖f x‖
[PROOFSTEP]
rw [memℒp_one_iff_integrable] at hf
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Integrable f
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ HasFiniteIntegral fun x => ‖f x‖
[PROOFSTEP]
exact hf.norm.2
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{x | n ≤ ‖f x‖₊}.indicator f a - 0‖ ∂μ) atTop (𝓝 0) :=
  by
  refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
  refine' fun n => univ_mem' (id fun x => _)
  by_cases hx : (n : ℝ) ≤ ‖f x‖
  · dsimp
    rwa [Set.indicator_of_mem]
  · dsimp
    rw [Set.indicator_of_not_mem, norm_zero]
    · exact norm_nonneg _
    · assumption
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ) atTop (𝓝 0)
[PROOFSTEP]
refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖
[PROOFSTEP]
refine' fun n => univ_mem' (id fun x => _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
⊢ x ∈ {x | (fun a => ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖) x}
[PROOFSTEP]
by_cases hx : (n : ℝ) ≤ ‖f x‖
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a => ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖) x}
[PROOFSTEP]
dsimp
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f x‖ ≤ ‖f x‖
[PROOFSTEP]
rwa [Set.indicator_of_mem]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a => ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖) x}
[PROOFSTEP]
dsimp
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f x‖ ≤ ‖f x‖
[PROOFSTEP]
rw [Set.indicator_of_not_mem, norm_zero]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ 0 ≤ ‖f x‖
[PROOFSTEP]
exact norm_nonneg _
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ¬↑n ≤ ‖f x‖
⊢ ¬x ∈ {x | ↑n ≤ ‖f x‖₊}
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this : Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ) atTop (𝓝 0)
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [ENNReal.tendsto_atTop_zero] at this
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hM⟩ := this (ENNReal.ofReal ε) (ENNReal.ofReal_pos.2 hε)
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
M : ℕ
hM : ∀ (n : ℕ), n ≥ M → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ENNReal.ofReal ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp only [true_and_iff, ge_iff_le, zero_tsub, zero_le, sub_zero, zero_add, coe_nnnorm, Set.mem_Icc] at hM
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
M : ℕ
hM : ∀ (n : ℕ), M ≤ n → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ∂μ ≤ ENNReal.ofReal ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨M, _⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
M : ℕ
hM : ∀ (n : ℕ), M ≤ n → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ∂μ ≤ ENNReal.ofReal ε
⊢ ∫⁻ (x : α), ↑‖Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
convert hM M le_rfl
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
M : ℕ
hM : ∀ (n : ℕ), M ≤ n → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ∂μ ≤ ENNReal.ofReal ε
x✝ : α
⊢ ↑‖Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x✝‖₊ = ENNReal.ofReal ‖Set.indicator {x | ↑M ≤ ‖f x‖₊} f x✝‖
[PROOFSTEP]
simp only [coe_nnnorm, ENNReal.ofReal_eq_coe_nnreal (norm_nonneg _)]
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
this :
  ∀ (ε : ℝ≥0∞),
    ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ ≤ ε
M : ℕ
hM : ∀ (n : ℕ), M ≤ n → ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ∂μ ≤ ENNReal.ofReal ε
x✝ : α
⊢ ↑‖Set.indicator {x | ↑M ≤ ‖f x‖} f x✝‖₊ =
    ↑{ val := ‖Set.indicator {x | ↑M ≤ ‖f x‖₊} f x✝‖, property := (_ : 0 ≤ ‖Set.indicator {x | ↑M ≤ ‖f x‖₊} f x✝‖) }
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
⊢ ∫⁻ (x : α), ↑‖Set.indicator {x | max M 0 ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simpa
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
⊢ ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hf_mk : Memℒp (hf.1.mk f) 1 μ := (memℒp_congr_ae hf.1.ae_eq_mk).mp hf
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
hf_mk : Memℒp (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) 1
⊢ ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hM_pos, hfM⟩ := hf_mk.integral_indicator_norm_ge_nonneg_le_of_meas μ hf.1.stronglyMeasurable_mk hε
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
hf_mk : Memℒp (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) 1
M : ℝ
hM_pos : 0 ≤ M
hfM :
  ∫⁻ (x : α),
      ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
            (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊ ∂μ ≤
    ENNReal.ofReal ε
⊢ ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨M, hM_pos, (le_of_eq _).trans hfM⟩
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
hf_mk : Memℒp (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) 1
M : ℝ
hM_pos : 0 ≤ M
hfM :
  ∫⁻ (x : α),
      ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
            (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊ ∂μ ≤
    ENNReal.ofReal ε
⊢ ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ =
    ∫⁻ (x : α),
      ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
            (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊ ∂μ
[PROOFSTEP]
refine' lintegral_congr_ae _
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
hf_mk : Memℒp (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) 1
M : ℝ
hM_pos : 0 ≤ M
hfM :
  ∫⁻ (x : α),
      ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
            (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊ ∂μ ≤
    ENNReal.ofReal ε
⊢ (fun x => ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊) =ᵐ[μ] fun x =>
    ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
          (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊
[PROOFSTEP]
filter_upwards [hf.1.ae_eq_mk] with x hx
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
ε : ℝ
hε : 0 < ε
hf_mk : Memℒp (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) 1
M : ℝ
hM_pos : 0 ≤ M
hfM :
  ∫⁻ (x : α),
      ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
            (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊ ∂μ ≤
    ENNReal.ofReal ε
x : α
hx : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
⊢ ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ =
    ↑‖Set.indicator {x | M ≤ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊}
          (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x‖₊
[PROOFSTEP]
simp only [Set.indicator_apply, coe_nnnorm, Set.mem_setOf_eq, ENNReal.coe_eq_coe, hx.symm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
⊢ ∃ M, snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0
[PROOFSTEP]
have hbdd : snormEssSup f μ < ∞ := hf.snorm_lt_top
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ ∃ M, snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0
[PROOFSTEP]
refine' ⟨(snorm f ∞ μ + 1).toReal, _⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ snormEssSup (Set.indicator {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} f) μ = 0
[PROOFSTEP]
rw [snormEssSup_indicator_eq_snormEssSup_restrict]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ snormEssSup f (Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}) = 0
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}
[PROOFSTEP]
have : μ.restrict {x : α | (snorm f ⊤ μ + 1).toReal ≤ ‖f x‖₊} = 0 :=
  by
  simp only [coe_nnnorm, snorm_exponent_top, Measure.restrict_eq_zero]
  have : {x : α | (snormEssSup f μ + 1).toReal ≤ ‖f x‖} ⊆ {x : α | snormEssSup f μ < ‖f x‖₊} :=
    by
    intro x hx
    rw [Set.mem_setOf_eq, ← ENNReal.toReal_lt_toReal hbdd.ne ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, coe_nnnorm]
    refine' lt_of_lt_of_le _ hx
    rw [ENNReal.toReal_lt_toReal hbdd.ne]
    · exact ENNReal.lt_add_right hbdd.ne one_ne_zero
    · exact (ENNReal.add_lt_top.2 ⟨hbdd, ENNReal.one_lt_top⟩).ne
  rw [← nonpos_iff_eq_zero]
  refine' (measure_mono this).trans _
  have hle := coe_nnnorm_ae_le_snormEssSup f μ
  simp_rw [ae_iff, not_le] at hle
  exact nonpos_iff_eq_zero.2 hle
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} = 0
[PROOFSTEP]
simp only [coe_nnnorm, snorm_exponent_top, Measure.restrict_eq_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} = 0
[PROOFSTEP]
have : {x : α | (snormEssSup f μ + 1).toReal ≤ ‖f x‖} ⊆ {x : α | snormEssSup f μ < ‖f x‖₊} :=
  by
  intro x hx
  rw [Set.mem_setOf_eq, ← ENNReal.toReal_lt_toReal hbdd.ne ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, coe_nnnorm]
  refine' lt_of_lt_of_le _ hx
  rw [ENNReal.toReal_lt_toReal hbdd.ne]
  · exact ENNReal.lt_add_right hbdd.ne one_ne_zero
  · exact (ENNReal.add_lt_top.2 ⟨hbdd, ENNReal.one_lt_top⟩).ne
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
[PROOFSTEP]
intro x hx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
x : α
hx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}
⊢ x ∈ {x | snormEssSup f μ < ↑‖f x‖₊}
[PROOFSTEP]
rw [Set.mem_setOf_eq, ← ENNReal.toReal_lt_toReal hbdd.ne ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, coe_nnnorm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
x : α
hx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}
⊢ ENNReal.toReal (snormEssSup f μ) < ‖f x‖
[PROOFSTEP]
refine' lt_of_lt_of_le _ hx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
x : α
hx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}
⊢ ENNReal.toReal (snormEssSup f μ) < ENNReal.toReal (snormEssSup f μ + 1)
[PROOFSTEP]
rw [ENNReal.toReal_lt_toReal hbdd.ne]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
x : α
hx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}
⊢ snormEssSup f μ < snormEssSup f μ + 1
[PROOFSTEP]
exact ENNReal.lt_add_right hbdd.ne one_ne_zero
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
x : α
hx : x ∈ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖}
⊢ snormEssSup f μ + 1 ≠ ⊤
[PROOFSTEP]
exact (ENNReal.add_lt_top.2 ⟨hbdd, ENNReal.one_lt_top⟩).ne
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} = 0
[PROOFSTEP]
rw [← nonpos_iff_eq_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
⊢ ↑↑μ {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ≤ 0
[PROOFSTEP]
refine' (measure_mono this).trans _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0
[PROOFSTEP]
have hle := coe_nnnorm_ae_le_snormEssSup f μ
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
hle : ∀ᵐ (x : α) ∂μ, ↑‖f x‖₊ ≤ snormEssSup f μ
⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0
[PROOFSTEP]
simp_rw [ae_iff, not_le] at hle
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : {x | ENNReal.toReal (snormEssSup f μ + 1) ≤ ‖f x‖} ⊆ {x | snormEssSup f μ < ↑‖f x‖₊}
hle : ↑↑μ {a | snormEssSup f μ < ↑‖f a‖₊} = 0
⊢ ↑↑μ {x | snormEssSup f μ < ↑‖f x‖₊} ≤ 0
[PROOFSTEP]
exact nonpos_iff_eq_zero.2 hle
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
this : Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊} = 0
⊢ snormEssSup f (Measure.restrict μ {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}) = 0
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}
[PROOFSTEP]
rw [this, snormEssSup_measure_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤
hmeas : StronglyMeasurable f
hbdd : snormEssSup f μ < ⊤
⊢ MeasurableSet {x | ENNReal.toReal (snorm f ⊤ μ + 1) ≤ ↑‖f x‖₊}
[PROOFSTEP]
exact measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hp_ne_zero : p = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨1, hp_ne_zero.symm ▸ _⟩
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ snorm (Set.indicator {x | 1 ≤ ↑‖f x‖₊} f) 0 μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp [snorm_exponent_zero]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hp_ne_top : p = ∞
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : p = ⊤
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
subst hp_ne_top
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hf : Memℒp f ⊤
hp_ne_zero : ¬⊤ = 0
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) ⊤ μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hM⟩ := hf.snormEssSup_indicator_norm_ge_eq_zero μ hmeas
[GOAL]
case pos.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hf : Memℒp f ⊤
hp_ne_zero : ¬⊤ = 0
M : ℝ
hM : snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) ⊤ μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨M, _⟩
[GOAL]
case pos.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hf : Memℒp f ⊤
hp_ne_zero : ¬⊤ = 0
M : ℝ
hM : snormEssSup (Set.indicator {x | M ≤ ↑‖f x‖₊} f) μ = 0
⊢ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) ⊤ μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp only [snorm_exponent_top, hM, zero_le]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hM', hM⟩ :=
  Memℒp.integral_indicator_norm_ge_nonneg_le (μ := μ) (hf.norm_rpow hp_ne_zero hp_ne_top)
    (Real.rpow_pos_of_pos hε p.toReal)
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
⊢ ∃ M, snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨M ^ (1 / p.toReal), _⟩
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
⊢ snorm (Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top, ← ENNReal.rpow_one (ENNReal.ofReal ε)]
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
⊢ (∫⁻ (x : α), ↑‖Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f x‖₊ ^ ENNReal.toReal p ∂μ) ^
      (1 / ENNReal.toReal p) ≤
    ENNReal.ofReal ε ^ 1
[PROOFSTEP]
conv_rhs => rw [← mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
| ENNReal.ofReal ε ^ 1
[PROOFSTEP]
rw [← mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
| ENNReal.ofReal ε ^ 1
[PROOFSTEP]
rw [← mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
| ENNReal.ofReal ε ^ 1
[PROOFSTEP]
rw [← mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
⊢ (∫⁻ (x : α), ↑‖Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f x‖₊ ^ ENNReal.toReal p ∂μ) ^
      (1 / ENNReal.toReal p) ≤
    ENNReal.ofReal ε ^ (ENNReal.toReal p * (1 / ENNReal.toReal p))
[PROOFSTEP]
rw [ENNReal.rpow_mul, ENNReal.rpow_le_rpow_iff (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),
  ENNReal.ofReal_rpow_of_pos hε]
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
⊢ ∫⁻ (x : α), ↑‖Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f x‖₊ ^ ENNReal.toReal p ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
[PROOFSTEP]
convert hM
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x✝ : α
⊢ ↑‖Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f x✝‖₊ ^ ENNReal.toReal p =
    ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x✝‖₊
[PROOFSTEP]
rename_i x
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
⊢ ↑‖Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} f x‖₊ ^ ENNReal.toReal p =
    ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊
[PROOFSTEP]
rw [ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,
  nnnorm_indicator_eq_indicator_nnnorm]
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
⊢ ↑(Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} (fun a => ‖f a‖₊) x ^ ENNReal.toReal p) =
    ↑(Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun a => ‖‖f a‖ ^ ENNReal.toReal p‖₊) x)
[PROOFSTEP]
have hiff : M ^ (1 / p.toReal) ≤ ‖f x‖₊ ↔ M ≤ ‖‖f x‖ ^ p.toReal‖₊ := by
  rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm, ←
    Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)
      (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),
    ← Real.rpow_mul (norm_nonneg _), mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm,
    Real.rpow_one]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
⊢ M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
[PROOFSTEP]
rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm, ←
  Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)
    (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),
  ← Real.rpow_mul (norm_nonneg _), mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one]
[GOAL]
case h.e'_3.h.e'_4.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
⊢ ↑(Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} (fun a => ‖f a‖₊) x ^ ENNReal.toReal p) =
    ↑(Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun a => ‖‖f a‖ ^ ENNReal.toReal p‖₊) x)
[PROOFSTEP]
by_cases hx : x ∈ {x : α | M ^ (1 / p.toReal) ≤ ‖f x‖₊}
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ↑(Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} (fun a => ‖f a‖₊) x ^ ENNReal.toReal p) =
    ↑(Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun a => ‖‖f a‖ ^ ENNReal.toReal p‖₊) x)
[PROOFSTEP]
rw [Set.indicator_of_mem hx, Set.indicator_of_mem, Real.nnnorm_of_nonneg]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ↑(‖f x‖₊ ^ ENNReal.toReal p) = ↑{ val := ‖f x‖ ^ ENNReal.toReal p, property := ?pos✝ }
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ 0 ≤ ‖f x‖ ^ ENNReal.toReal p
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ x ∈ {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊}
[PROOFSTEP]
rfl
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ x ∈ {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊}
[PROOFSTEP]
rw [Set.mem_setOf_eq]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
[PROOFSTEP]
rwa [← hiff]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : ¬x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ↑(Set.indicator {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊} (fun a => ‖f a‖₊) x ^ ENNReal.toReal p) =
    ↑(Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun a => ‖‖f a‖ ^ ENNReal.toReal p‖₊) x)
[PROOFSTEP]
rw [Set.indicator_of_not_mem hx, Set.indicator_of_not_mem]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : ¬x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ↑(0 ^ ENNReal.toReal p) = ↑0
[PROOFSTEP]
simp [(ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : ¬x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ¬x ∈ {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊}
[PROOFSTEP]
rw [Set.mem_setOf_eq]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊} (fun x => ‖f x‖ ^ ENNReal.toReal p) x‖₊ ∂μ ≤
    ENNReal.ofReal (ε ^ ENNReal.toReal p)
x : α
hiff : M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
hx : ¬x ∈ {x | M ^ (1 / ENNReal.toReal p) ≤ ↑‖f x‖₊}
⊢ ¬M ≤ ↑‖‖f x‖ ^ ENNReal.toReal p‖₊
[PROOFSTEP]
rwa [← hiff]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ M, 0 < M ∧ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hM⟩ := hf.snorm_indicator_norm_ge_le μ hmeas hε
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ∃ M, 0 < M ∧ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _), le_trans (snorm_mono fun x => _) hM⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
⊢ ‖Set.indicator {x | max M 1 ≤ ↑‖f x‖₊} f x‖ ≤ ‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖
[PROOFSTEP]
rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm]
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
⊢ Set.indicator {x | max M 1 ≤ ↑‖f x‖₊} (fun a => ‖f a‖) x ≤ Set.indicator {x | M ≤ ↑‖f x‖₊} (fun a => ‖f a‖) x
[PROOFSTEP]
refine' Set.indicator_le_indicator_of_subset (fun x hx => _) (fun x => norm_nonneg (f x)) x
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x✝ x : α
hx : x ∈ {x | max M 1 ≤ ↑‖f x‖₊}
⊢ x ∈ {x | M ≤ ↑‖f x‖₊}
[PROOFSTEP]
rw [Set.mem_setOf_eq] at hx
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x✝ x : α
hx : max M 1 ≤ ↑‖f x‖₊
⊢ x ∈ {x | M ≤ ↑‖f x‖₊}
[PROOFSTEP]
exact (max_le_iff.1 hx).1
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hM : M ≤ 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨1, zero_lt_one, fun s _ _ => _⟩
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑μ s ≤ ENNReal.ofReal 1
⊢ snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [(_ : f = 0)]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑μ s ≤ ENNReal.ofReal 1
⊢ snorm (indicator s 0) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp [hε.le]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑μ s ≤ ENNReal.ofReal 1
⊢ f = 0
[PROOFSTEP]
ext x
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑μ s ≤ ENNReal.ofReal 1
x : α
⊢ f x = OfNat.ofNat 0 x
[PROOFSTEP]
rw [Pi.zero_apply, ← norm_le_zero_iff]
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : M ≤ 0
s : Set α
x✝¹ : MeasurableSet s
x✝ : ↑↑μ s ≤ ENNReal.ofReal 1
x : α
⊢ ‖f x‖ ≤ 0
[PROOFSTEP]
exact (lt_of_lt_of_le (hf x) hM).le
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : ¬M ≤ 0
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [not_le] at hM
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨(ε / M) ^ p.toReal, Real.rpow_pos_of_pos (div_pos hε hM) _, fun s hs hμ => _⟩
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
⊢ snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hp : p = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : p = 0
⊢ snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp [hp]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
⊢ snorm (indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [snorm_indicator_eq_snorm_restrict hs]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε
[PROOFSTEP]
have haebdd : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ M := by
  filter_upwards
  exact fun x => (hf x).le
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
[PROOFSTEP]
filter_upwards
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
⊢ ∀ (a : α), ‖f a‖ ≤ M
[PROOFSTEP]
exact fun x => (hf x).le
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ snorm f p (Measure.restrict μ s) ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' le_trans (snorm_le_of_ae_bound haebdd) _
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ ↑↑(Measure.restrict μ s) univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal M ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [Measure.restrict_apply MeasurableSet.univ, Set.univ_inter, ←
  ENNReal.le_div_iff_mul_le (Or.inl _) (Or.inl ENNReal.ofReal_ne_top)]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ ↑↑μ s ^ (ENNReal.toReal p)⁻¹ ≤ ENNReal.ofReal ε / ENNReal.ofReal M
[PROOFSTEP]
rw [← one_div, ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hp hp_top)]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ ↑↑μ s ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p
[PROOFSTEP]
refine' le_trans hμ _
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p) ≤ (ENNReal.ofReal ε / ENNReal.ofReal M) ^ ENNReal.toReal p
[PROOFSTEP]
rw [← ENNReal.ofReal_rpow_of_pos (div_pos hε hM), ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp hp_top),
  ENNReal.ofReal_div_of_pos hM]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
M : ℝ
hf : ∀ (x : α), ‖f x‖ < M
hM : 0 < M
s : Set α
hs : MeasurableSet s
hμ : ↑↑μ s ≤ ENNReal.ofReal ((ε / M) ^ ENNReal.toReal p)
hp : ¬p = 0
haebdd : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖f x‖ ≤ M
⊢ ENNReal.ofReal M ≠ 0
[PROOFSTEP]
simpa only [ENNReal.ofReal_eq_zero, not_le, Ne.def]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ δ hδ,
    ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨M, hMpos, hM⟩ := hf.snorm_indicator_norm_ge_pos_le μ hmeas hε
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ hδ,
    ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ :=
  snorm_indicator_le_of_bound μ (f := {x | ‖f x‖ < M}.indicator f) hp_top hε
    (by
      intro x
      rw [norm_indicator_eq_indicator_norm, Set.indicator_apply]
      split_ifs with h
      exacts [h, hMpos])
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ∀ (x : α), ‖Set.indicator {x | ‖f x‖ < M} f x‖ < ?m.135727
[PROOFSTEP]
intro x
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
⊢ ‖Set.indicator {x | ‖f x‖ < M} f x‖ < ?m.135727
[PROOFSTEP]
rw [norm_indicator_eq_indicator_norm, Set.indicator_apply]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
⊢ (if x ∈ {x | ‖f x‖ < M} then ‖f x‖ else 0) < ?m.135727
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ℝ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ℝ
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
h : x ∈ {x | ‖f x‖ < M}
⊢ ‖f x‖ < ?m.135727
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
x : α
h : ¬x ∈ {x | ‖f x‖ < M}
⊢ 0 < ?m.135727
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ℝ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
⊢ ℝ
[PROOFSTEP]
exacts [h, hMpos]
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ hδ,
    ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδpos, fun s hs hμs => _⟩
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal ε
[PROOFSTEP]
rw [(_ : f = {x : α | M ≤ ‖f x‖₊}.indicator f + {x : α | ‖f x‖ < M}.indicator f)]
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator s (Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f)) p μ ≤
    2 * ENNReal.ofReal ε
[PROOFSTEP]
rw [snorm_indicator_eq_snorm_restrict hs]
[GOAL]
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f) p (Measure.restrict μ s) ≤
    2 * ENNReal.ofReal ε
[PROOFSTEP]
refine' le_trans (snorm_add_le _ _ hp_one) _
[GOAL]
case intro.intro.intro.intro.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ AEStronglyMeasurable (Set.indicator {x | M ≤ ↑‖f x‖₊} f) (Measure.restrict μ s)
[PROOFSTEP]
exact
  StronglyMeasurable.aestronglyMeasurable
    (hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe))
[GOAL]
case intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ AEStronglyMeasurable (Set.indicator {x | ‖f x‖ < M} f) (Measure.restrict μ s)
[PROOFSTEP]
exact
  StronglyMeasurable.aestronglyMeasurable
    (hmeas.indicator (measurableSet_lt hmeas.nnnorm.measurable.subtype_coe measurable_const))
[GOAL]
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p (Measure.restrict μ s) +
      snorm (Set.indicator {x | ‖f x‖ < M} f) p (Measure.restrict μ s) ≤
    2 * ENNReal.ofReal ε
[PROOFSTEP]
rw [two_mul]
[GOAL]
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p (Measure.restrict μ s) +
      snorm (Set.indicator {x | ‖f x‖ < M} f) p (Measure.restrict μ s) ≤
    ENNReal.ofReal ε + ENNReal.ofReal ε
[PROOFSTEP]
refine' add_le_add (le_trans (snorm_mono_measure _ Measure.restrict_le_self) hM) _
[GOAL]
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator {x | ‖f x‖ < M} f) p (Measure.restrict μ s) ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [← snorm_indicator_eq_snorm_restrict hs]
[GOAL]
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact hδ s hs hμs
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ f = Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f
[PROOFSTEP]
ext x
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
⊢ f x = (Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f) x
[PROOFSTEP]
by_cases hx : M ≤ ‖f x‖
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : M ≤ ‖f x‖
⊢ f x = (Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f) x
[PROOFSTEP]
rw [Pi.add_apply, Set.indicator_of_mem, Set.indicator_of_not_mem, add_zero]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : M ≤ ‖f x‖
⊢ ¬x ∈ {x | ‖f x‖ < M}
[PROOFSTEP]
simpa
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : M ≤ ‖f x‖
⊢ x ∈ {x | M ≤ ↑‖f x‖₊}
[PROOFSTEP]
simpa
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : ¬M ≤ ‖f x‖
⊢ f x = (Set.indicator {x | M ≤ ↑‖f x‖₊} f + Set.indicator {x | ‖f x‖ < M} f) x
[PROOFSTEP]
rw [Pi.add_apply, Set.indicator_of_not_mem, Set.indicator_of_mem, zero_add]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : ¬M ≤ ‖f x‖
⊢ x ∈ {x | ‖f x‖ < M}
[PROOFSTEP]
simpa using hx
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : snorm (Set.indicator {x | M ≤ ↑‖f x‖₊} f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s (Set.indicator {x | ‖f x‖ < M} f)) p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
x : α
hx : ¬M ≤ ‖f x‖
⊢ ¬x ∈ {x | M ≤ ↑‖f x‖₊}
[PROOFSTEP]
simpa using hx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ := hf.snorm_indicator_le' μ hp_one hp_top hmeas (half_pos hε)
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal (ε / 2)
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδpos, fun s hs hμs => le_trans (hδ s hs hμs) _⟩
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal (ε / 2)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ 2 * ENNReal.ofReal (ε / 2) ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [ENNReal.ofReal_div_of_pos zero_lt_two, (by norm_num : ENNReal.ofReal 2 = 2), ENNReal.mul_div_cancel']
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal (ε / 2)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ ENNReal.ofReal 2 = 2
[PROOFSTEP]
norm_num
[GOAL]
case intro.intro.h0
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal (ε / 2)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ 2 ≠ 0
[PROOFSTEP]
norm_num
[GOAL]
case intro.intro.hI
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ 2 * ENNReal.ofReal (ε / 2)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ 2 ≠ ⊤
[PROOFSTEP]
norm_num
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
ε : ℝ
hε : 0 < ε
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hℒp := hf
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : Memℒp f p
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨⟨f', hf', heq⟩, _⟩ := hf
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
right✝ : snorm f p μ < ⊤
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ := (hℒp.ae_eq heq).snorm_indicator_le_of_meas μ hp_one hp_top hf' hε
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
right✝ : snorm f p μ < ⊤
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f') p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ hδ, ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδpos, fun s hs hμs => _⟩
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
right✝ : snorm f p μ < ⊤
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f') p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator s f) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
convert hδ s hs hμs using 1
[GOAL]
case h.e'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
right✝ : snorm f p μ < ⊤
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f') p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (Set.indicator s f) p μ = snorm (Set.indicator s f') p μ
[PROOFSTEP]
rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs]
[GOAL]
case h.e'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
ε : ℝ
hε : 0 < ε
hℒp : Memℒp f p
right✝ : snorm f p μ < ⊤
f' : α → β
hf' : StronglyMeasurable f'
heq : f =ᵐ[μ] f'
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s f') p μ ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm f p (Measure.restrict μ s) = snorm f' p (Measure.restrict μ s)
[PROOFSTEP]
refine' snorm_congr_ae heq.restrict
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : α → β
hp : 1 ≤ p
hp_ne_top : p ≠ ⊤
hg : Memℒp g p
⊢ UnifIntegrable (fun x => g) p μ
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : α → β
hp : 1 ≤ p
hp_ne_top : p ≠ ⊤
hg : Memℒp g p
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδ_pos, hgδ⟩ := hg.snorm_indicator_le μ hp hp_ne_top hε
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : α → β
hp : 1 ≤ p
hp_ne_top : p ≠ ⊤
hg : Memℒp g p
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hgδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s g) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact ⟨δ, hδ_pos, fun _ => hgδ⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
⊢ UnifIntegrable f p μ
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hι : Nonempty ι
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
cases' hι with i
[GOAL]
case pos.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
i : ι
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ := (hf i).snorm_indicator_le μ hp_one hp_top hε
[GOAL]
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
i : ι
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδpos, fun j s hs hμs => _⟩
[GOAL]
case pos.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
i : ι
δ : ℝ
hδpos : 0 < δ
hδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
j : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (indicator s (f j)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
convert hδ s hs hμs
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact ⟨1, zero_lt_one, fun i => False.elim <| hι <| Nonempty.intro i⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
f : Fin n → α → β
hf : ∀ (i : Fin n), Memℒp (f i) p
⊢ UnifIntegrable f p μ
[PROOFSTEP]
revert f
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
⊢ ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
[PROOFSTEP]
induction' n with n h
[GOAL]
case zero
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
⊢ ∀ {f : Fin Nat.zero → α → β}, (∀ (i : Fin Nat.zero), Memℒp (f i) p) → UnifIntegrable f p μ
[PROOFSTEP]
intro f hf
[GOAL]
case zero
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : Fin Nat.zero → α → β
hf : ∀ (i : Fin Nat.zero), Memℒp (f i) p
⊢ UnifIntegrable f p μ
[PROOFSTEP]
have : Subsingleton (Fin Nat.zero) := subsingleton_fin_zero
[GOAL]
case zero
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : Fin Nat.zero → α → β
hf : ∀ (i : Fin Nat.zero), Memℒp (f i) p
this : Subsingleton (Fin Nat.zero)
⊢ UnifIntegrable f p μ
[PROOFSTEP]
exact unifIntegrable_subsingleton μ hp_one hp_top hf
[GOAL]
case succ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
⊢ ∀ {f : Fin (Nat.succ n) → α → β}, (∀ (i : Fin (Nat.succ n)), Memℒp (f i) p) → UnifIntegrable f p μ
[PROOFSTEP]
intro f hfLp ε hε
[GOAL]
case succ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : Fin (Nat.succ n)) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
let g : Fin n → α → β := fun k => f k
[GOAL]
case succ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
⊢ ∃ δ x,
    ∀ (i : Fin (Nat.succ n)) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hgLp : ∀ i, Memℒp (g i) p μ := fun i => hfLp i
[GOAL]
case succ
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
⊢ ∃ δ x,
    ∀ (i : Fin (Nat.succ n)) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ₁, hδ₁pos, hδ₁⟩ := h hgLp hε
[GOAL]
case succ.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : Fin (Nat.succ n)) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ₂, hδ₂pos, hδ₂⟩ := (hfLp n).snorm_indicator_le μ hp_one hp_top hε
[GOAL]
case succ.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : Fin (Nat.succ n)) (s : Set α),
      MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨min δ₁ δ₂, lt_min hδ₁pos hδ₂pos, fun i s hs hμs => _⟩
[GOAL]
case succ.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hi : i.val < n
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [(_ : f i = g ⟨i.val, hi⟩)]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ snorm (indicator s (g { val := ↑i, isLt := hi })) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact hδ₁ _ s hs (le_trans hμs <| ENNReal.ofReal_le_ofReal <| min_le_left _ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ f i = g { val := ↑i, isLt := hi }
[PROOFSTEP]
simp
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ¬↑i < n
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [(_ : i = n)]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ¬↑i < n
⊢ snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact hδ₂ _ hs (le_trans hμs <| ENNReal.ofReal_le_ofReal <| min_le_right _ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ¬↑i < n
⊢ i = ↑n
[PROOFSTEP]
have hi' := Fin.is_lt i
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ¬↑i < n
hi' : ↑i < Nat.succ n
⊢ i = ↑n
[PROOFSTEP]
rw [Nat.lt_succ_iff] at hi'
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ¬↑i < n
hi' : ↑i ≤ n
⊢ i = ↑n
[PROOFSTEP]
rw [not_lt] at hi
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : n ≤ ↑i
hi' : ↑i ≤ n
⊢ i = ↑n
[PROOFSTEP]
ext
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : n ≤ ↑i
hi' : ↑i ≤ n
⊢ ↑i = ↑↑n
[PROOFSTEP]
symm
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p) → UnifIntegrable f p μ
f : Fin (Nat.succ n) → α → β
hfLp : ∀ (i : Fin (Nat.succ n)), Memℒp (f i) p
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k => f ↑↑k
hgLp : ∀ (i : Fin n), Memℒp (g i) p
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s (f ↑n)) p μ ≤ ENNReal.ofReal ε
i : Fin (Nat.succ n)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : n ≤ ↑i
hi' : ↑i ≤ n
⊢ ↑↑n = ↑i
[PROOFSTEP]
rw [Fin.coe_ofNat_eq_mod, le_antisymm hi' hi, Nat.mod_succ_eq_iff_lt, Nat.lt_succ]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
⊢ UnifIntegrable f p μ
[PROOFSTEP]
obtain ⟨n, hn⟩ := Finite.exists_equiv_fin ι
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
⊢ UnifIntegrable f p μ
[PROOFSTEP]
intro ε hε
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
set g : Fin n → α → β := f ∘ hn.some.symm with hgeq
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hg : ∀ i, Memℒp (g i) p μ := fun _ => hf _
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
hg : ∀ (i : Fin n), Memℒp (g i) p
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ := unifIntegrable_fin μ hp_one hp_top hg hε
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
hg : ∀ (i : Fin n), Memℒp (g i) p
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδpos, fun i s hs hμs => _⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
hg : ∀ (i : Fin n), Memℒp (g i) p
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s ((fun i => g i) i)) p μ ≤ ENNReal.ofReal ε
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
specialize hδ (hn.some i) s hs hμs
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
hg : ∀ (i : Fin n), Memℒp (g i) p
δ : ℝ
hδpos : 0 < δ
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
hδ : snorm (indicator s ((fun i => g i) (↑(Nonempty.some hn) i))) p μ ≤ ENNReal.ofReal ε
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp_rw [hgeq, Function.comp_apply, Equiv.symm_apply_apply] at hδ
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), Memℒp (f i) p
n : ℕ
hn : Nonempty (ι ≃ Fin n)
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := f ∘ ↑(Nonempty.some hn).symm
hgeq : g = f ∘ ↑(Nonempty.some hn).symm
hg : ∀ (i : Fin n), Memℒp (g i) p
δ : ℝ
hδpos : 0 < δ
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ
hδ : snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
assumption
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
⊢ snorm (indicator s (f - g)) p μ ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
by_cases hp : p = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : p = 0
⊢ snorm (indicator s (f - g)) p μ ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
simp [hp]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
⊢ snorm (indicator s (f - g)) p μ ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
have : ∀ x, ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun _ => c) x‖ :=
  by
  intro x
  by_cases hx : x ∈ s
  · rw [Set.indicator_of_mem hx, Set.indicator_of_mem hx, Pi.sub_apply, ← dist_eq_norm, Real.norm_eq_abs,
      abs_of_nonneg hc]
    exact hf x hx
  · simp [Set.indicator_of_not_mem hx]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
⊢ ∀ (x : α), ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
[PROOFSTEP]
intro x
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
x : α
⊢ ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
[PROOFSTEP]
by_cases hx : x ∈ s
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
x : α
hx : x ∈ s
⊢ ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
[PROOFSTEP]
rw [Set.indicator_of_mem hx, Set.indicator_of_mem hx, Pi.sub_apply, ← dist_eq_norm, Real.norm_eq_abs, abs_of_nonneg hc]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
x : α
hx : x ∈ s
⊢ dist (f x) (g x) ≤ c
[PROOFSTEP]
exact hf x hx
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
x : α
hx : ¬x ∈ s
⊢ ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
[PROOFSTEP]
simp [Set.indicator_of_not_mem hx]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
⊢ snorm (indicator s (f - g)) p μ ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' le_trans (snorm_mono this) _
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
⊢ snorm (fun x => indicator s (fun x => c) x) p μ ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
rw [snorm_indicator_const hs hp hp']
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
⊢ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ≤ ENNReal.ofReal c * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' mul_le_mul_right' (le_of_eq _) _
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p✝ p : ℝ≥0∞
hp' : p ≠ ⊤
s : Set α
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ (x : α), x ∈ s → dist (f x) (g x) ≤ c
hp : ¬p = 0
this : ∀ (x : α), ‖indicator s (f - g) x‖ ≤ ‖indicator s (fun x => c) x‖
⊢ ↑‖c‖₊ = ENNReal.ofReal c
[PROOFSTEP]
rw [← ofReal_norm_eq_coe_nnnorm, Real.norm_eq_abs, abs_of_nonneg hc]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
rw [ENNReal.tendsto_atTop_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
by_cases ε < ∞
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
by_cases ε < ∞
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ¬ε < ⊤
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
swap
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ¬ε < ⊤
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
rw [not_lt, top_le_iff] at h
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε = ⊤
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
exact ⟨0, fun n _ => by simp [h]⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε = ⊤
n : ℕ
x✝ : n ≥ 0
⊢ snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
simp [h]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
by_cases hμ : μ = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : μ = 0
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
exact ⟨0, fun n _ => by simp [hμ]⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : μ = 0
n : ℕ
x✝ : n ≥ 0
⊢ snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
simp [hμ]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
have hε' : 0 < ε.toReal / 3 := div_pos (ENNReal.toReal_pos (gt_iff_lt.1 hε).ne.symm h.ne) (by norm_num)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
⊢ 0 < 3
[PROOFSTEP]
norm_num
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
have hdivp : 0 ≤ 1 / p.toReal := by
  refine' one_div_nonneg.2 _
  rw [← ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']
  exact le_trans (zero_le _) hp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
⊢ 0 ≤ 1 / ENNReal.toReal p
[PROOFSTEP]
refine' one_div_nonneg.2 _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
⊢ 0 ≤ ENNReal.toReal p
[PROOFSTEP]
rw [← ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
⊢ 0 ≤ p
[PROOFSTEP]
exact le_trans (zero_le _) hp
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
have hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal) := Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
obtain ⟨δ₁, hδ₁, hsnorm₁⟩ := hui hε'
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
obtain ⟨δ₂, hδ₂, hsnorm₂⟩ := hg'.snorm_indicator_le μ hp hp' hε'
[GOAL]
case neg.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
obtain ⟨t, htm, ht₁, ht₂⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg (lt_min hδ₁ hδ₂)
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
ht₂ : TendstoUniformlyOn (fun n => f n) g atTop tᶜ
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
rw [Metric.tendstoUniformlyOn_iff] at ht₂
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
ht₂ : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
specialize
  ht₂ (ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
    (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 hε).ne.symm h.ne) (mul_pos (by norm_num) hpow))
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
ht₂ : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε
⊢ 0 < 3
[PROOFSTEP]
norm_num
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
ht₂ :
  ∀ᶠ (n : ℕ) in atTop,
    ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
obtain ⟨N, hN⟩ := eventually_atTop.1 ht₂
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
ht₂ :
  ∀ᶠ (n : ℕ) in atTop,
    ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
clear ht₂
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
refine' ⟨N, fun n hn => _⟩
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (f n - g) p μ ≤ ε
[PROOFSTEP]
rw [← t.indicator_self_add_compl (f n - g)]
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (indicator t (f n - g) + indicator tᶜ (f n - g)) p μ ≤ ε
[PROOFSTEP]
refine'
  le_trans
    (snorm_add_le (((hf n).sub hg).indicator htm).aestronglyMeasurable
      (((hf n).sub hg).indicator htm.compl).aestronglyMeasurable hp)
    _
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (indicator t (f n - g)) p μ + snorm (indicator tᶜ (f n - g)) p μ ≤ ε
[PROOFSTEP]
rw [sub_eq_add_neg, Set.indicator_add' t, Set.indicator_neg']
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (indicator t (f n) + -indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
refine'
  le_trans
    (add_le_add_right
      (snorm_add_le ((hf n).indicator htm).aestronglyMeasurable (hg.indicator htm).neg.aestronglyMeasurable hp) _)
    _
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
have hnf : snorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3) :=
  by
  refine' hsnorm₁ n t htm (le_trans ht₁ _)
  rw [ENNReal.ofReal_le_ofReal_iff hδ₁.le]
  exact min_le_left _ _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
refine' hsnorm₁ n t htm (le_trans ht₁ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ ENNReal.ofReal (min δ₁ δ₂) ≤ ENNReal.ofReal δ₁
[PROOFSTEP]
rw [ENNReal.ofReal_le_ofReal_iff hδ₁.le]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
⊢ min δ₁ δ₂ ≤ δ₁
[PROOFSTEP]
exact min_le_left _ _
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
have hng : snorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3) :=
  by
  refine' hsnorm₂ t htm (le_trans ht₁ _)
  rw [ENNReal.ofReal_le_ofReal_iff hδ₂.le]
  exact min_le_right _ _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
refine' hsnorm₂ t htm (le_trans ht₁ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ ENNReal.ofReal (min δ₁ δ₂) ≤ ENNReal.ofReal δ₂
[PROOFSTEP]
rw [ENNReal.ofReal_le_ofReal_iff hδ₂.le]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ min δ₁ δ₂ ≤ δ₂
[PROOFSTEP]
exact min_le_right _ _
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
have hlt : snorm (tᶜ.indicator (f n - g)) p μ ≤ ENNReal.ofReal (ε.toReal / 3) :=
  by
  specialize hN n hn
  have : 0 ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)) :=
    by
    rw [div_mul_eq_div_mul_one_div]
    exact mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
  have :=
    snorm_sub_le_of_dist_bdd μ hp' htm.compl this fun x hx =>
      (dist_comm (g x) (f n x) ▸ (hN x hx).le :
        dist (f n x) (g x) ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
  refine' le_trans this _
  rw [div_mul_eq_div_mul_one_div, ← ENNReal.ofReal_toReal (measure_lt_top μ tᶜ).ne,
    ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, ← ENNReal.ofReal_mul, mul_assoc]
  · refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right hε'.le _)
    rw [mul_comm, mul_one_div, div_le_one]
    · refine'
        Real.rpow_le_rpow ENNReal.toReal_nonneg (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos hμ).le _) hdivp
      rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]
      exact measure_mono (Set.subset_univ _)
    · exact Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _
  · refine' mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
specialize hN n hn
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
have : 0 ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)) :=
  by
  rw [div_mul_eq_div_mul_one_div]
  exact mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
[PROOFSTEP]
rw [div_mul_eq_div_mul_one_div]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ 0 ≤ ENNReal.toReal ε / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
[PROOFSTEP]
exact mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
⊢ snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
have :=
  snorm_sub_le_of_dist_bdd μ hp' htm.compl this fun x hx =>
    (dist_comm (g x) (f n x) ▸ (hN x hx).le :
      dist (f n x) (g x) ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)))
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
refine' le_trans this _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p) ≤
    ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
rw [div_mul_eq_div_mul_one_div, ← ENNReal.ofReal_toReal (measure_lt_top μ tᶜ).ne,
  ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, ← ENNReal.ofReal_mul, mul_assoc]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ENNReal.ofReal
      (ENNReal.toReal ε / 3 *
        (1 / ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (↑↑μ tᶜ) ^ (1 / ENNReal.toReal p))) ≤
    ENNReal.ofReal (ENNReal.toReal ε / 3)
[PROOFSTEP]
refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right hε'.le _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ 1 / ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (↑↑μ tᶜ) ^ (1 / ENNReal.toReal p) ≤ 1
[PROOFSTEP]
rw [mul_comm, mul_one_div, div_le_one]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ENNReal.toReal (↑↑μ tᶜ) ^ (1 / ENNReal.toReal p) ≤ ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' Real.rpow_le_rpow ENNReal.toReal_nonneg (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos hμ).le _) hdivp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ↑↑μ tᶜ ≤ ENNReal.ofReal ↑(measureUnivNNReal μ)
[PROOFSTEP]
rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ ↑↑μ tᶜ ≤ ↑↑μ univ
[PROOFSTEP]
exact measure_mono (Set.subset_univ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
exact Real.rpow_pos_of_pos (measureUnivNNReal_pos hμ) _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hN : ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this✝ : 0 ≤ ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
this :
  snorm (indicator tᶜ ((fun x => f n x) - fun x => g x)) p μ ≤
    ENNReal.ofReal (ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))) *
      ↑↑μ tᶜ ^ (1 / ENNReal.toReal p)
⊢ 0 ≤ ENNReal.toReal ε / 3 * (1 / ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
[PROOFSTEP]
refine' mul_nonneg hε'.le (one_div_nonneg.2 hpow.le)
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
have : ENNReal.ofReal (ε.toReal / 3) = ε / 3 :=
  by
  rw [ENNReal.ofReal_div_of_pos (show (0 : ℝ) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]
  simp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ ENNReal.ofReal (ENNReal.toReal ε / 3) = ε / 3
[PROOFSTEP]
rw [ENNReal.ofReal_div_of_pos (show (0 : ℝ) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ 0 < 3
[PROOFSTEP]
norm_num
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
⊢ ε / ENNReal.ofReal 3 = ε / 3
[PROOFSTEP]
simp
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hng : snorm (indicator t g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
this : ENNReal.ofReal (ENNReal.toReal ε / 3) = ε / 3
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
rw [this] at hnf hng hlt
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ε / 3
hng : snorm (indicator t g) p μ ≤ ε / 3
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ε / 3
this : ENNReal.ofReal (ENNReal.toReal ε / 3) = ε / 3
⊢ snorm (indicator t (f n)) p μ + snorm (-indicator t g) p μ + snorm (indicator tᶜ (f n + -g)) p μ ≤ ε
[PROOFSTEP]
rw [snorm_neg, ← ENNReal.add_thirds ε, ← sub_eq_add_neg]
[GOAL]
case neg.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ⊤
hμ : ¬μ = 0
hε' : 0 < ENNReal.toReal ε / 3
hdivp : 0 ≤ 1 / ENNReal.toReal p
hpow : 0 < ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p)
δ₁ : ℝ
hδ₁ : 0 < δ₁
hsnorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
hsnorm₂ :
  ∀ (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₂ → snorm (indicator s g) p μ ≤ ENNReal.ofReal (ENNReal.toReal ε / 3)
t : Set α
htm : MeasurableSet t
ht₁ : ↑↑μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N : ℕ
hN :
  ∀ (b : ℕ),
    b ≥ N →
      ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ENNReal.toReal ε / (3 * ↑(measureUnivNNReal μ) ^ (1 / ENNReal.toReal p))
n : ℕ
hn : n ≥ N
hnf : snorm (indicator t (f n)) p μ ≤ ε / 3
hng : snorm (indicator t g) p μ ≤ ε / 3
hlt : snorm (indicator tᶜ (f n - g)) p μ ≤ ε / 3
this : ENNReal.ofReal (ENNReal.toReal ε / 3) = ε / 3
⊢ snorm (indicator t (f n)) p μ + snorm (indicator t g) p μ + snorm (indicator tᶜ (f n - g)) p μ ≤ ε / 3 + ε / 3 + ε / 3
[PROOFSTEP]
exact add_le_add_three hnf hng hlt
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
have : ∀ n, snorm (f n - g) p μ = snorm ((hf n).mk (f n) - hg.1.mk g) p μ := fun n =>
  snorm_congr_ae ((hf n).ae_eq_mk.sub hg.1.ae_eq_mk)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
⊢ Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
simp_rw [this]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
⊢ Tendsto
    (fun n =>
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ)
    atTop (𝓝 0)
[PROOFSTEP]
refine'
  tendsto_Lp_of_tendsto_ae_of_meas μ hp hp' (fun n => (hf n).stronglyMeasurable_mk) hg.1.stronglyMeasurable_mk
    (hg.ae_eq hg.1.ae_eq_mk) (hui.ae_eq fun n => (hf n).ae_eq_mk) _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
⊢ ∀ᵐ (x : α) ∂μ,
    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x) atTop
      (𝓝 (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x))
[PROOFSTEP]
have h_ae_forall_eq : ∀ᵐ x ∂μ, ∀ n, f n x = (hf n).mk (f n) x :=
  by
  rw [ae_all_iff]
  exact fun n => (hf n).ae_eq_mk
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
⊢ ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
[PROOFSTEP]
rw [ae_all_iff]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
⊢ ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, f i a = AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) μ) a
[PROOFSTEP]
exact fun n => (hf n).ae_eq_mk
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
⊢ ∀ᵐ (x : α) ∂μ,
    Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x) atTop
      (𝓝 (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x))
[PROOFSTEP]
filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
x : α
hx_tendsto : Tendsto (fun n => f n x) atTop (𝓝 (g x))
hxf_eq : ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x
⊢ Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x) atTop
    (𝓝 (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x))
[PROOFSTEP]
rw [← hxg_eq]
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
x : α
hx_tendsto : Tendsto (fun n => f n x) atTop (𝓝 (g x))
hxf_eq : ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x
⊢ Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x) atTop (𝓝 (g x))
[PROOFSTEP]
convert hx_tendsto using 1
[GOAL]
case h.e'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
x : α
hx_tendsto : Tendsto (fun n => f n x) atTop (𝓝 (g x))
hxf_eq : ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x
⊢ (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x) = fun n => f n x
[PROOFSTEP]
ext1 n
[GOAL]
case h.e'_3.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
this :
  ∀ (n : ℕ),
    snorm (f n - g) p μ =
      snorm
        (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) -
          AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ))
        p μ
h_ae_forall_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
x : α
hx_tendsto : Tendsto (fun n => f n x) atTop (𝓝 (g x))
hxf_eq : ∀ (n : ℕ), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x
hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g μ) x
n : ℕ
⊢ AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) μ) x = f n x
[PROOFSTEP]
exact (hxf_eq n).symm
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : Tendsto (fun n => snorm (f n) p μ) atTop (𝓝 0)
⊢ UnifIntegrable f p μ
[PROOFSTEP]
intro ε hε
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : Tendsto (fun n => snorm (f n) p μ) atTop (𝓝 0)
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [ENNReal.tendsto_atTop_zero] at hf_tendsto
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨N, hN⟩ := hf_tendsto (ENNReal.ofReal ε) (by simpa)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
⊢ ENNReal.ofReal ε > 0
[PROOFSTEP]
simpa
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
let F : Fin N → α → β := fun n => f n
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hF : ∀ n, Memℒp (F n) p μ := fun n => hf n
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ₁, hδpos₁, hδ₁⟩ := unifIntegrable_fin μ hp hp' hF hε
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => F i) i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ₁, hδpos₁, fun n s hs hμs => _⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => F i) i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ₁
⊢ snorm (indicator s (f n)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hn : n < N
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => F i) i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ₁
hn : n < N
⊢ snorm (indicator s (f n)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact hδ₁ ⟨n, hn⟩ s hs hμs
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hf_tendsto : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ε
ε : ℝ
hε : 0 < ε
N : ℕ
hN : ∀ (n : ℕ), n ≥ N → snorm (f n) p μ ≤ ENNReal.ofReal ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p
δ₁ : ℝ
hδpos₁ : 0 < δ₁
hδ₁ :
  ∀ (i : Fin N) (s : Set α),
    MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ₁ → snorm (indicator s ((fun i => F i) i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal δ₁
hn : ¬n < N
⊢ snorm (indicator s (f n)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact (snorm_indicator_le _).trans (hN n (not_lt.1 hn))
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
⊢ UnifIntegrable f p μ
[PROOFSTEP]
have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
⊢ f = (fun x => g) + fun n => f n - g
[PROOFSTEP]
ext1 n
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
n : ℕ
⊢ f n = ((fun x => g) + fun n => f n - g) n
[PROOFSTEP]
simp
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x => g) + fun n => f n - g
⊢ UnifIntegrable f p μ
[PROOFSTEP]
rw [this]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x => g) + fun n => f n - g
⊢ UnifIntegrable ((fun x => g) + fun n => f n - g) p μ
[PROOFSTEP]
refine' UnifIntegrable.add _ _ hp (fun _ => hg.aestronglyMeasurable) fun n => (hf n).1.sub hg.aestronglyMeasurable
[GOAL]
case refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x => g) + fun n => f n - g
⊢ UnifIntegrable (fun x => g) p μ
[PROOFSTEP]
exact unifIntegrable_const μ hp hp' hg
[GOAL]
case refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p
hg : Memℒp g p
hfg : Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x => g) + fun n => f n - g
⊢ UnifIntegrable (fun n => f n - g) p μ
[PROOFSTEP]
exact unifIntegrable_of_tendsto_Lp_zero μ hp hp' (fun n => (hf n).sub hg) hfg
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
⊢ Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
refine' tendsto_of_subseq_tendsto fun ns hns => _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
obtain ⟨ms, _, hms'⟩ := TendstoInMeasure.exists_seq_tendsto_ae fun ε hε => (hfg ε hε).comp hns
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : Memℒp g p
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ℕ → ℕ
hns : Tendsto ns atTop atTop
ms : ℕ → ℕ
left✝ : StrictMono ms
hms' : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns (ms i)) x) atTop (𝓝 (g x))
⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)
[PROOFSTEP]
exact
  ⟨ms,
    tendsto_Lp_of_tendsto_ae μ hp hp' (fun _ => hf _) hg
      (fun ε hε =>
        let ⟨δ, hδ, hδ'⟩ := hui hε
        ⟨δ, hδ, fun i s hs hμs => hδ' _ s hs hμs⟩)
      hms'⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ UnifIntegrable f p μ
[PROOFSTEP]
have hpzero := (lt_of_lt_of_le zero_lt_one hp).ne.symm
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
⊢ UnifIntegrable f p μ
[PROOFSTEP]
by_cases hμ : μ Set.univ = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ↑↑μ univ = 0
⊢ UnifIntegrable f p μ
[PROOFSTEP]
rw [Measure.measure_univ_eq_zero] at hμ
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : μ = 0
⊢ UnifIntegrable f p μ
[PROOFSTEP]
exact hμ.symm ▸ unifIntegrable_zero_meas
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
⊢ UnifIntegrable f p μ
[PROOFSTEP]
intro ε hε
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨C, hCpos, hC⟩ := h (ε / 2) (half_pos hε)
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
⊢ ∃ δ x,
    ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine'
  ⟨(ε / (2 * C)) ^ ENNReal.toReal p, Real.rpow_pos_of_pos (div_pos hε (mul_pos two_pos (NNReal.coe_pos.2 hCpos))) _,
    fun i s hs hμs => _⟩
[GOAL]
case neg.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hμs' : μ s = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ↑↑μ s = 0
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [(snorm_eq_zero_iff ((hf i).indicator hs).aestronglyMeasurable hpzero).2 (indicator_meas_zero hμs')]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ↑↑μ s = 0
⊢ 0 ≤ ENNReal.ofReal ε
[PROOFSTEP]
norm_num
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
calc
  snorm (Set.indicator s (f i)) p μ ≤
      snorm (Set.indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i)) p μ +
        snorm (Set.indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ :=
    by
    refine'
      le_trans (Eq.le _)
        (snorm_add_le
          (StronglyMeasurable.aestronglyMeasurable
            ((hf i).indicator (hs.inter (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))))
          (StronglyMeasurable.aestronglyMeasurable
            ((hf i).indicator (hs.inter ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const))))
          hp)
    congr
    change _ = fun x => (s ∩ {x : α | C ≤ ‖f i x‖₊}).indicator (f i) x + (s ∩ {x : α | ‖f i x‖₊ < C}).indicator (f i) x
    rw [← Set.indicator_union_of_disjoint]
    · congr
      rw [← Set.inter_union_distrib_left,
        (by ext; simp [le_or_lt] : {x : α | C ≤ ‖f i x‖₊} ∪ {x : α | ‖f i x‖₊ < C} = Set.univ), Set.inter_univ]
    · refine' (Disjoint.inf_right' _ _).inf_left' _
      rw [disjoint_iff_inf_le]
      rintro x ⟨hx₁, hx₂⟩
      rw [Set.mem_setOf_eq] at hx₁ hx₂
      exact False.elim (hx₂.ne (eq_of_le_of_not_lt hx₁ (not_lt.2 hx₂.le)).symm)
  _ ≤ snorm (Set.indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ + (C : ℝ≥0∞) * μ s ^ (1 / ENNReal.toReal p) :=
    by
    refine' add_le_add (snorm_mono fun x => norm_indicator_le_of_subset (Set.inter_subset_right _ _) _ _) _
    rw [← Set.indicator_indicator]
    rw [snorm_indicator_eq_snorm_restrict hs]
    have : ∀ᵐ x ∂μ.restrict s, ‖{x : α | ‖f i x‖₊ < C}.indicator (f i) x‖ ≤ C :=
      by
      refine' ae_of_all _ _
      simp_rw [norm_indicator_eq_indicator_norm]
      exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _
    refine' le_trans (snorm_le_of_ae_bound this) _
    rw [mul_comm, Measure.restrict_apply' hs, Set.univ_inter, ENNReal.ofReal_coe_nnreal, one_div]
  _ ≤ ENNReal.ofReal (ε / 2) + C * ENNReal.ofReal (ε / (2 * C)) :=
    by
    refine' add_le_add (hC i) (mul_le_mul_left' _ _)
    rwa [ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hpzero hp'),
      ENNReal.ofReal_rpow_of_pos (div_pos hε (mul_pos two_pos (NNReal.coe_pos.2 hCpos)))]
  _ ≤ ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) :=
    by
    refine' add_le_add_left _ _
    rw [← ENNReal.ofReal_coe_nnreal, ← ENNReal.ofReal_mul (NNReal.coe_nonneg _), ← div_div,
      mul_div_cancel' _ (NNReal.coe_pos.2 hCpos).ne.symm]
  _ ≤ ENNReal.ofReal ε := by rw [← ENNReal.ofReal_add (half_pos hε).le (half_pos hε).le, add_halves]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator s (f i)) p μ ≤
    snorm (indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i)) p μ + snorm (indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ
[PROOFSTEP]
refine'
  le_trans (Eq.le _)
    (snorm_add_le
      (StronglyMeasurable.aestronglyMeasurable
        ((hf i).indicator (hs.inter (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))))
      (StronglyMeasurable.aestronglyMeasurable
        ((hf i).indicator (hs.inter ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const))))
      hp)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator s (f i)) p μ =
    snorm (indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i) + indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ
[PROOFSTEP]
congr
[GOAL]
case e_f
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ indicator s (f i) = indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i) + indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)
[PROOFSTEP]
change _ = fun x => (s ∩ {x : α | C ≤ ‖f i x‖₊}).indicator (f i) x + (s ∩ {x : α | ‖f i x‖₊ < C}).indicator (f i) x
[GOAL]
case e_f
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ indicator s (f i) = fun x => indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i) x + indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i) x
[PROOFSTEP]
rw [← Set.indicator_union_of_disjoint]
[GOAL]
case e_f
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ indicator s (f i) = indicator (s ∩ {x | C ≤ ‖f i x‖₊} ∪ s ∩ {x | ‖f i x‖₊ < C}) (f i)
[PROOFSTEP]
congr
[GOAL]
case e_f
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ indicator s (f i) = indicator (s ∩ {x | C ≤ ‖f i x‖₊} ∪ s ∩ {x | ‖f i x‖₊ < C}) (f i)
[PROOFSTEP]
rw [← Set.inter_union_distrib_left,
  (by ext; simp [le_or_lt] : {x : α | C ≤ ‖f i x‖₊} ∪ {x : α | ‖f i x‖₊ < C} = Set.univ), Set.inter_univ]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ {x | C ≤ ‖f i x‖₊} ∪ {x | ‖f i x‖₊ < C} = univ
[PROOFSTEP]
ext
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
x✝ : α
⊢ x✝ ∈ {x | C ≤ ‖f i x‖₊} ∪ {x | ‖f i x‖₊ < C} ↔ x✝ ∈ univ
[PROOFSTEP]
simp [le_or_lt]
[GOAL]
case e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ Disjoint (s ∩ {x | C ≤ ‖f i x‖₊}) (s ∩ {x | ‖f i x‖₊ < C})
[PROOFSTEP]
refine' (Disjoint.inf_right' _ _).inf_left' _
[GOAL]
case e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ Disjoint {x | C ≤ ‖f i x‖₊} {x | ‖f i x‖₊ < C}
[PROOFSTEP]
rw [disjoint_iff_inf_le]
[GOAL]
case e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ {x | C ≤ ‖f i x‖₊} ⊓ {x | ‖f i x‖₊ < C} ≤ ⊥
[PROOFSTEP]
rintro x ⟨hx₁, hx₂⟩
[GOAL]
case e_f.h.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
x : α
hx₁ : x ∈ {x | C ≤ ‖f i x‖₊}
hx₂ : x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∈ ⊥
[PROOFSTEP]
rw [Set.mem_setOf_eq] at hx₁ hx₂
[GOAL]
case e_f.h.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
x : α
hx₁ : C ≤ ‖f i x‖₊
hx₂ : ‖f i x‖₊ < C
⊢ x ∈ ⊥
[PROOFSTEP]
exact False.elim (hx₂.ne (eq_of_le_of_not_lt hx₁ (not_lt.2 hx₂.le)).symm)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator (s ∩ {x | C ≤ ‖f i x‖₊}) (f i)) p μ + snorm (indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ ≤
    snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ + ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' add_le_add (snorm_mono fun x => norm_indicator_le_of_subset (Set.inter_subset_right _ _) _ _) _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator (s ∩ {x | ‖f i x‖₊ < C}) (f i)) p μ ≤ ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
rw [← Set.indicator_indicator]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator s (indicator {x | ‖f i x‖₊ < C} (f i))) p μ ≤ ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
rw [snorm_indicator_eq_snorm_restrict hs]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator {x | ‖f i x‖₊ < C} (f i)) p (Measure.restrict μ s) ≤ ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
have : ∀ᵐ x ∂μ.restrict s, ‖{x : α | ‖f i x‖₊ < C}.indicator (f i) x‖ ≤ C :=
  by
  refine' ae_of_all _ _
  simp_rw [norm_indicator_eq_indicator_norm]
  exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖ ≤ ↑C
[PROOFSTEP]
refine' ae_of_all _ _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ∀ (a : α), ‖indicator {x | ‖f i x‖₊ < C} (f i) a‖ ≤ ↑C
[PROOFSTEP]
simp_rw [norm_indicator_eq_indicator_norm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ∀ (a : α), indicator {x | ‖f i x‖₊ < C} (fun a => ‖f i a‖) a ≤ ↑C
[PROOFSTEP]
exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
this : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖ ≤ ↑C
⊢ snorm (indicator {x | ‖f i x‖₊ < C} (f i)) p (Measure.restrict μ s) ≤ ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' le_trans (snorm_le_of_ae_bound this) _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
this : ∀ᵐ (x : α) ∂Measure.restrict μ s, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖ ≤ ↑C
⊢ ↑↑(Measure.restrict μ s) univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal ↑C ≤ ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
rw [mul_comm, Measure.restrict_apply' hs, Set.univ_inter, ENNReal.ofReal_coe_nnreal, one_div]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ + ↑C * ↑↑μ s ^ (1 / ENNReal.toReal p) ≤
    ENNReal.ofReal (ε / 2) + ↑C * ENNReal.ofReal (ε / (2 * ↑C))
[PROOFSTEP]
refine' add_le_add (hC i) (mul_le_mul_left' _ _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ↑↑μ s ^ (1 / ENNReal.toReal p) ≤ ENNReal.ofReal (ε / (2 * ↑C))
[PROOFSTEP]
rwa [ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hpzero hp'),
  ENNReal.ofReal_rpow_of_pos (div_pos hε (mul_pos two_pos (NNReal.coe_pos.2 hCpos)))]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ENNReal.ofReal (ε / 2) + ↑C * ENNReal.ofReal (ε / (2 * ↑C)) ≤ ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)
[PROOFSTEP]
refine' add_le_add_left _ _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ↑C * ENNReal.ofReal (ε / (2 * ↑C)) ≤ ENNReal.ofReal (ε / 2)
[PROOFSTEP]
rw [← ENNReal.ofReal_coe_nnreal, ← ENNReal.ofReal_mul (NNReal.coe_nonneg _), ← div_div,
  mul_div_cancel' _ (NNReal.coe_pos.2 hCpos).ne.symm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬↑↑μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ ENNReal.toReal p)
hμs' : ¬↑↑μ s = 0
⊢ ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [← ENNReal.ofReal_add (half_pos hε).le (half_pos hε).le, add_halves]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ UnifIntegrable f p μ
[PROOFSTEP]
set g : ι → α → β := fun i => (hf i).choose
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
⊢ UnifIntegrable f p μ
[PROOFSTEP]
refine'
  (unifIntegrable_of' μ hp hp' (fun i => (Exists.choose_spec <| hf i).1) fun ε hε => _).ae_eq fun i =>
    (Exists.choose_spec <| hf i).2.symm
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
⊢ ∃ C,
    0 < C ∧
      ∀ (i : ι),
        snorm
            (indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
              (Exists.choose (_ : AEStronglyMeasurable (f i) μ)))
            p μ ≤
          ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨C, hC⟩ := h ε hε
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C,
    0 < C ∧
      ∀ (i : ι),
        snorm
            (indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
              (Exists.choose (_ : AEStronglyMeasurable (f i) μ)))
            p μ ≤
          ENNReal.ofReal ε
[PROOFSTEP]
have hCg : ∀ i, snorm ({x | C ≤ ‖g i x‖₊}.indicator (g i)) p μ ≤ ENNReal.ofReal ε :=
  by
  intro i
  refine' le_trans (le_of_eq <| snorm_congr_ae _) (hC i)
  filter_upwards [(Exists.choose_spec <| hf i).2] with x hx
  by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊}
  · rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]
    rwa [Set.mem_setOf, hx] at hfx
  · rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]
    rwa [Set.mem_setOf, hx] at hfx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
intro i
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' le_trans (le_of_eq <| snorm_congr_ae _) (hC i)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ indicator {x | C ≤ ‖g i x‖₊} (g i) =ᵐ[μ] indicator {x | C ≤ ‖f i x‖₊} (f i)
[PROOFSTEP]
filter_upwards [(Exists.choose_spec <| hf i).2] with x hx
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊}
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ ¬x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hCg : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C,
    0 < C ∧
      ∀ (i : ι),
        snorm
            (indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
              (Exists.choose (_ : AEStronglyMeasurable (f i) μ)))
            p μ ≤
          ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨max C 1, lt_max_of_lt_right one_pos, fun i => le_trans (snorm_mono fun x => _) (hCg i)⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hCg : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ ‖indicator {x | max C 1 ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
        (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x‖ ≤
    ‖indicator {x | C ≤ ‖g i x‖₊} (g i) x‖
[PROOFSTEP]
rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm]
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ℕ → α → β
g✝ : α → β
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
hCg : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ indicator {x | max C 1 ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (fun a => ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) a‖) x ≤
    indicator {x | C ≤ ‖g i x‖₊} (fun a => ‖g i a‖) x
[PROOFSTEP]
exact
  Set.indicator_le_indicator_of_subset (fun x hx => Set.mem_setOf_eq ▸ le_trans (le_max_left _ _) hx)
    (fun _ => norm_nonneg _) _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : ι → α → β
hf : UniformIntegrable f p μ
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
⊢ UniformIntegrable g p μ
[PROOFSTEP]
obtain ⟨hfm, hunif, C, hC⟩ := hf
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : ι → α → β
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
hfm : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hunif : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (f i) p μ ≤ ↑C
⊢ UniformIntegrable g p μ
[PROOFSTEP]
refine' ⟨fun i => (hfm i).congr (hfg i), (unifIntegrable_congr_ae hfg).1 hunif, C, fun i => _⟩
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : ι → α → β
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
hfm : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hunif : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (f i) p μ ≤ ↑C
i : ι
⊢ snorm (g i) p μ ≤ ↑C
[PROOFSTEP]
rw [← snorm_congr_ae (hfg i)]
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f g : ι → α → β
hfg : ∀ (n : ι), f n =ᵐ[μ] g n
hfm : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hunif : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (f i) p μ ≤ ↑C
i : ι
⊢ snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
exact hC i
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : ∀ (i : ι), Memℒp (f i) p
⊢ UniformIntegrable f p μ
[PROOFSTEP]
cases nonempty_fintype ι
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : ∀ (i : ι), Memℒp (f i) p
val✝ : Fintype ι
⊢ UniformIntegrable f p μ
[PROOFSTEP]
refine' ⟨fun n => (hf n).1, unifIntegrable_finite μ hp_one hp_top hf, _⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : ∀ (i : ι), Memℒp (f i) p
val✝ : Fintype ι
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
by_cases hι : Nonempty ι
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : ∀ (i : ι), Memℒp (f i) p
val✝ : Fintype ι
hι : Nonempty ι
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
choose _ hf using hf
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
set C :=
  (Finset.univ.image fun i : ι => snorm (f i) p μ).max'
    ⟨snorm (f hι.some) p μ, Finset.mem_image.2 ⟨hι.some, Finset.mem_univ _, rfl⟩⟩
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
refine' ⟨C.toNNReal, fun i => _⟩
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i : ι
⊢ snorm (f i) p μ ≤ ↑(ENNReal.toNNReal C)
[PROOFSTEP]
rw [ENNReal.coe_toNNReal]
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i : ι
⊢ snorm (f i) p μ ≤ C
[PROOFSTEP]
exact Finset.le_max' (α := ℝ≥0∞) _ _ (Finset.mem_image.2 ⟨i, Finset.mem_univ _, rfl⟩)
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i : ι
⊢ C ≠ ⊤
[PROOFSTEP]
refine' ne_of_lt ((Finset.max'_lt_iff _ _).2 fun y hy => _)
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i : ι
y : ℝ≥0∞
hy : y ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ
⊢ y < ⊤
[PROOFSTEP]
rw [Finset.mem_image] at hy
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i : ι
y : ℝ≥0∞
hy : ∃ a, a ∈ Finset.univ ∧ snorm (f a) p μ = y
⊢ y < ⊤
[PROOFSTEP]
obtain ⟨i, -, rfl⟩ := hy
[GOAL]
case pos.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
val✝ : Fintype ι
hι : Nonempty ι
h✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf : ∀ (i : ι), snorm (f i) p μ < ⊤
C : ℝ≥0∞ :=
  Finset.max' (Finset.image (fun i => snorm (f i) p μ) Finset.univ)
    (_ : ∃ x, x ∈ Finset.image (fun i => snorm (f i) p μ) Finset.univ)
i✝ i : ι
⊢ snorm (f i) p μ < ⊤
[PROOFSTEP]
exact hf i
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : Finite ι
hp_one : 1 ≤ p
hp_top : p ≠ ⊤
hf : ∀ (i : ι), Memℒp (f i) p
val✝ : Fintype ι
hι : ¬Nonempty ι
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
exact ⟨0, fun i => False.elim <| hι <| Nonempty.intro i⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ UniformIntegrable f p μ
[PROOFSTEP]
refine' ⟨fun i => (hf i).aestronglyMeasurable, unifIntegrable_of μ hp hp' (fun i => (hf i).aestronglyMeasurable) h, _⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
obtain ⟨C, hC⟩ := h 1 one_pos
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
⊢ ∃ C, ∀ (i : ι), snorm (f i) p μ ≤ ↑C
[PROOFSTEP]
refine' ⟨((C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1).toNNReal, fun i => _⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ snorm (f i) p μ ≤ ↑(ENNReal.toNNReal (↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1))
[PROOFSTEP]
calc
  snorm (f i) p μ ≤
      snorm ({x : α | ‖f i x‖₊ < C}.indicator (f i)) p μ + snorm ({x : α | C ≤ ‖f i x‖₊}.indicator (f i)) p μ :=
    by
    refine'
      le_trans (snorm_mono fun x => _)
        (snorm_add_le
          (StronglyMeasurable.aestronglyMeasurable
            ((hf i).indicator ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const)))
          (StronglyMeasurable.aestronglyMeasurable
            ((hf i).indicator (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm)))
          hp)
    · rw [Pi.add_apply, Set.indicator_apply]
      split_ifs with hx
      · rw [Set.indicator_of_not_mem, add_zero]
        simpa using hx
      · rw [Set.indicator_of_mem, zero_add]
        simpa using hx
  _ ≤ (C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1 :=
    by
    have : ∀ᵐ x ∂μ, ‖{x : α | ‖f i x‖₊ < C}.indicator (f i) x‖₊ ≤ C :=
      by
      refine' eventually_of_forall _
      simp_rw [nnnorm_indicator_eq_indicator_nnnorm]
      exact Set.indicator_le fun x (hx : _ < _) => hx.le
    refine' add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ENNReal.ofReal_one ▸ hC i)
    simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm]
    exact le_rfl
  _ = ((C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1 : ℝ≥0∞).toNNReal :=
    by
    rw [ENNReal.coe_toNNReal]
    exact
      ENNReal.add_ne_top.2
        ⟨ENNReal.mul_ne_top ENNReal.coe_ne_top
            (ENNReal.rpow_ne_top_of_nonneg (inv_nonneg.2 ENNReal.toReal_nonneg) (measure_lt_top _ _).ne),
          ENNReal.one_ne_top⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ snorm (f i) p μ ≤ snorm (indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ
[PROOFSTEP]
refine'
  le_trans (snorm_mono fun x => _)
    (snorm_add_le
      (StronglyMeasurable.aestronglyMeasurable
        ((hf i).indicator ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const)))
      (StronglyMeasurable.aestronglyMeasurable
        ((hf i).indicator (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm)))
      hp)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
⊢ ‖f i x‖ ≤ ‖(indicator {x | ‖f i x‖₊ < C} (f i) + indicator {x | C ≤ ‖f i x‖₊} (f i)) x‖
[PROOFSTEP]
rw [Pi.add_apply, Set.indicator_apply]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
⊢ ‖f i x‖ ≤ ‖(if x ∈ {x | ‖f i x‖₊ < C} then f i x else 0) + indicator {x | C ≤ ‖f i x‖₊} (f i) x‖
[PROOFSTEP]
split_ifs with hx
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ ‖f i x‖ ≤ ‖f i x + indicator {x | C ≤ ‖f i x‖₊} (f i) x‖
[PROOFSTEP]
rw [Set.indicator_of_not_mem, add_zero]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ ¬x ∈ {x | C ≤ ‖f i x‖₊}
[PROOFSTEP]
simpa using hx
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : ¬x ∈ {x | ‖f i x‖₊ < C}
⊢ ‖f i x‖ ≤ ‖0 + indicator {x | C ≤ ‖f i x‖₊} (f i) x‖
[PROOFSTEP]
rw [Set.indicator_of_mem, zero_add]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : ¬x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∈ {x | C ≤ ‖f i x‖₊}
[PROOFSTEP]
simpa using hx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ snorm (indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤
    ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1
[PROOFSTEP]
have : ∀ᵐ x ∂μ, ‖{x : α | ‖f i x‖₊ < C}.indicator (f i) x‖₊ ≤ C :=
  by
  refine' eventually_of_forall _
  simp_rw [nnnorm_indicator_eq_indicator_nnnorm]
  exact Set.indicator_le fun x (hx : _ < _) => hx.le
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ ∀ᵐ (x : α) ∂μ, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C
[PROOFSTEP]
refine' eventually_of_forall _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ ∀ (x : α), ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C
[PROOFSTEP]
simp_rw [nnnorm_indicator_eq_indicator_nnnorm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ ∀ (x : α), indicator {x | ‖f i x‖₊ < C} (fun a => ‖f i a‖₊) x ≤ C
[PROOFSTEP]
exact Set.indicator_le fun x (hx : _ < _) => hx.le
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
this : ∀ᵐ (x : α) ∂μ, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C
⊢ snorm (indicator {x | ‖f i x‖₊ < C} (f i)) p μ + snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤
    ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1
[PROOFSTEP]
refine' add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ENNReal.ofReal_one ▸ hC i)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
this : ∀ᵐ (x : α) ∂μ, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C
⊢ ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ * ENNReal.ofReal ((fun a => ↑a) C) ≤ ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹
[PROOFSTEP]
simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
this : ∀ᵐ (x : α) ∂μ, ‖indicator {x | ‖f i x‖₊ < C} (f i) x‖₊ ≤ C
⊢ ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ ≤ ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹
[PROOFSTEP]
exact le_rfl
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1 = ↑(ENNReal.toNNReal (↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1))
[PROOFSTEP]
rw [ENNReal.coe_toNNReal]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
⊢ ↑C * ↑↑μ univ ^ (ENNReal.toReal p)⁻¹ + 1 ≠ ⊤
[PROOFSTEP]
exact
  ENNReal.add_ne_top.2
    ⟨ENNReal.mul_ne_top ENNReal.coe_ne_top
        (ENNReal.rpow_ne_top_of_nonneg (inv_nonneg.2 ENNReal.toReal_nonneg) (measure_lt_top _ _).ne),
      ENNReal.one_ne_top⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ UniformIntegrable f p μ
[PROOFSTEP]
set g : ι → α → β := fun i => (hf i).choose
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
⊢ UniformIntegrable f p μ
[PROOFSTEP]
have hgmeas : ∀ i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hf i).1
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
⊢ UniformIntegrable f p μ
[PROOFSTEP]
have hgeq : ∀ i, g i =ᵐ[μ] f i := fun i => (Exists.choose_spec <| hf i).2.symm
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
⊢ UniformIntegrable f p μ
[PROOFSTEP]
refine' (uniformIntegrable_of' hp hp' hgmeas fun ε hε => _).ae_eq hgeq
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨C, hC⟩ := h ε hε
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨C, fun i => le_trans (le_of_eq <| snorm_congr_ae _) (hC i)⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ indicator {x | C ≤ ‖g i x‖₊} (g i) =ᵐ[μ] indicator {x | C ≤ ‖f i x‖₊} (f i)
[PROOFSTEP]
filter_upwards [(Exists.choose_spec <| hf i).2] with x hx
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊}
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x =
    indicator {x | C ≤ ‖f i x‖₊} (f i) x
[PROOFSTEP]
rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgeq : ∀ (i : ι), g i =ᵐ[μ] f i
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ ¬x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨-, hfu, M, hM⟩ := hfu
[GOAL]
case intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
[GOAL]
case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ {x | C ≤ ‖f i x‖₊} ≤ ENNReal.ofReal δ :=
  by
  by_contra hcon; push_neg at hcon
  choose ℐ hℐ using hcon
  lift δ to ℝ≥0 using hδpos.le
  have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ snorm (f (ℐ C)) p μ :=
    by
    intro C
    calc
      C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ C • μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.toReal) :=
        by
        rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
        simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
        refine' mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))
      _ ≤ snorm ({x | C ≤ ‖f (ℐ C) x‖₊}.indicator (f (ℐ C))) p μ :=
        by
        refine'
          snorm_indicator_ge_of_bdd_below hp hp' _ (measurableSet_le measurable_const (hf _).nnnorm.measurable)
            (eventually_of_forall fun x hx => _)
        rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]
      _ ≤ snorm (f (ℐ C)) p μ := snorm_indicator_le _
  specialize this (2 * max M 1 * HPow.hPow δ⁻¹ (1 / p.toReal))
  rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), ← ENNReal.coe_smul, smul_eq_mul, mul_assoc,
    NNReal.inv_rpow, inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 hδpos)).ne.symm, mul_one, ENNReal.coe_mul, ←
    NNReal.inv_rpow] at this
  refine'
    (lt_of_le_of_lt (le_trans (hM <| ℐ <| 2 * max M 1 * HPow.hPow δ⁻¹ (1 / p.toReal)) (le_max_left (M : ℝ≥0∞) 1))
          (lt_of_lt_of_le _ this)).ne
      rfl
  rw [← ENNReal.coe_one, ← ENNReal.coe_max, ← ENNReal.coe_mul, ENNReal.coe_lt_coe]
  exact lt_two_mul_self (lt_max_of_lt_right one_pos)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), ↑↑μ {x | C ≤ ‖f i x‖₊} ≤ ENNReal.ofReal δ
[PROOFSTEP]
by_contra hcon
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hcon : ¬∃ C, ∀ (i : ι), ↑↑μ {x | C ≤ ‖f i x‖₊} ≤ ENNReal.ofReal δ
⊢ False
[PROOFSTEP]
push_neg at hcon
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hcon : ∀ (C : ℝ≥0), ∃ i, ENNReal.ofReal δ < ↑↑μ {x | C ≤ ‖f i x‖₊}
⊢ False
[PROOFSTEP]
choose ℐ hℐ using hcon
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
ℐ : ℝ≥0 → ι
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
⊢ False
[PROOFSTEP]
lift δ to ℝ≥0 using hδpos.le
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
⊢ False
[PROOFSTEP]
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ snorm (f (ℐ C)) p μ :=
  by
  intro C
  calc
    C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ C • μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.toReal) :=
      by
      rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
      simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
      refine' mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))
    _ ≤ snorm ({x | C ≤ ‖f (ℐ C) x‖₊}.indicator (f (ℐ C))) p μ :=
      by
      refine'
        snorm_indicator_ge_of_bdd_below hp hp' _ (measurableSet_le measurable_const (hf _).nnnorm.measurable)
          (eventually_of_forall fun x hx => _)
      rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]
    _ ≤ snorm (f (ℐ C)) p μ := snorm_indicator_le _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
⊢ ∀ (C : ℝ≥0), C • ↑δ ^ (1 / ENNReal.toReal p) ≤ snorm (f (ℐ C)) p μ
[PROOFSTEP]
intro C
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ C • ↑δ ^ (1 / ENNReal.toReal p) ≤ snorm (f (ℐ C)) p μ
[PROOFSTEP]
calc
  C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ C • μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / p.toReal) :=
    by
    rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
    simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
    refine' mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))
  _ ≤ snorm ({x | C ≤ ‖f (ℐ C) x‖₊}.indicator (f (ℐ C))) p μ :=
    by
    refine'
      snorm_indicator_ge_of_bdd_below hp hp' _ (measurableSet_le measurable_const (hf _).nnnorm.measurable)
        (eventually_of_forall fun x hx => _)
    rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]
  _ ≤ snorm (f (ℐ C)) p μ := snorm_indicator_le _
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ C • ↑δ ^ (1 / ENNReal.toReal p) ≤ C • ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ ↑C * ↑δ ^ (1 / ENNReal.toReal p) ≤ ↑C * ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hℐ : ∀ (C : ℝ≥0), ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
⊢ ↑C * ↑δ ^ (1 / ENNReal.toReal p) ≤ ↑C * ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / ENNReal.toReal p)
[PROOFSTEP]
refine' mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
⊢ C • ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊} ^ (1 / ENNReal.toReal p) ≤ snorm (indicator {x | C ≤ ‖f (ℐ C) x‖₊} (f (ℐ C))) p μ
[PROOFSTEP]
refine'
  snorm_indicator_ge_of_bdd_below hp hp' _ (measurableSet_le measurable_const (hf _).nnnorm.measurable)
    (eventually_of_forall fun x hx => _)
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
C : ℝ≥0
x : α
hx : x ∈ {x | C ≤ ‖f (ℐ C) x‖₊}
⊢ C ≤ ‖indicator {x | C ≤ ‖f (ℐ C) x‖₊} (f (ℐ C)) x‖₊
[PROOFSTEP]
rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ∀ (C : ℝ≥0), C • ↑δ ^ (1 / ENNReal.toReal p) ≤ snorm (f (ℐ C)) p μ
⊢ False
[PROOFSTEP]
specialize this (2 * max M 1 * HPow.hPow δ⁻¹ (1 / p.toReal))
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
this :
  (2 * max M 1 * δ⁻¹ ^ (1 / ENNReal.toReal p)) • ↑δ ^ (1 / ENNReal.toReal p) ≤
    snorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / ENNReal.toReal p)))) p μ
⊢ False
[PROOFSTEP]
rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), ← ENNReal.coe_smul, smul_eq_mul, mul_assoc,
  NNReal.inv_rpow, inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 hδpos)).ne.symm, mul_one, ENNReal.coe_mul, ←
  NNReal.inv_rpow] at this
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ↑2 * ↑(max M 1) ≤ snorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / ENNReal.toReal p)))) p μ
⊢ False
[PROOFSTEP]
refine'
  (lt_of_le_of_lt (le_trans (hM <| ℐ <| 2 * max M 1 * HPow.hPow δ⁻¹ (1 / p.toReal)) (le_max_left (M : ℝ≥0∞) 1))
        (lt_of_lt_of_le _ this)).ne
    rfl
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ↑2 * ↑(max M 1) ≤ snorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / ENNReal.toReal p)))) p μ
⊢ max (↑M) 1 < ↑2 * ↑(max M 1)
[PROOFSTEP]
rw [← ENNReal.coe_one, ← ENNReal.coe_max, ← ENNReal.coe_mul, ENNReal.coe_lt_coe]
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal ↑δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < ↑↑μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ↑2 * ↑(max M 1) ≤ snorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / ENNReal.toReal p)))) p μ
⊢ max M 1 < 2 * max M 1
[PROOFSTEP]
exact lt_two_mul_self (lt_max_of_lt_right one_pos)
[GOAL]
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), snorm (f i) p μ ≤ ↑M
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), ↑↑μ {x | C ≤ ‖f i x‖₊} ≤ ENNReal.ofReal δ
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
exact ⟨C, fun i => hδ i _ (measurableSet_le measurable_const (hf i).nnnorm.measurable) (hC i)⟩
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
set g : ι → α → β := fun i => (hfu.1 i).choose
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hgmeas : ∀ i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hfu.1 i).1
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have hgunif : UniformIntegrable g p μ := hfu.ae_eq fun i => (Exists.choose_spec <| hfu.1 i).2
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨C, hC⟩ := hgunif.spec' hp hp' hgmeas hε
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ C, ∀ (i : ι), snorm (indicator {x | C ≤ ‖f i x‖₊} (f i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨C, fun i => le_trans (le_of_eq <| snorm_congr_ae _) (hC i)⟩
[GOAL]
case intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
⊢ indicator {x | C ≤ ‖f i x‖₊} (f i) =ᵐ[μ] indicator {x | C ≤ ‖g i x‖₊} (g i)
[PROOFSTEP]
filter_upwards [(Exists.choose_spec <| hfu.1 i).2] with x hx
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
⊢ indicator {x | C ≤ ‖f i x‖₊} (f i) x =
    indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x
[PROOFSTEP]
by_cases hfx : x ∈ {x | C ≤ ‖f i x‖₊}
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖f i x‖₊} (f i) x =
    indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x
[PROOFSTEP]
rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx]
[GOAL]
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : x ∈ {x | C ≤ ‖f i x‖₊}
⊢ x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ indicator {x | C ≤ ‖f i x‖₊} (f i) x =
    indicator {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
      (Exists.choose (_ : AEStronglyMeasurable (f i) μ)) x
[PROOFSTEP]
rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]
[GOAL]
case neg.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ⊤
hfu : UniformIntegrable f p μ
ε : ℝ
hε : 0 < ε
g : ι → α → β := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) μ)
hgmeas : ∀ (i : ι), StronglyMeasurable (g i)
hgunif : UniformIntegrable g p μ
C : ℝ≥0
hC : ∀ (i : ι), snorm (indicator {x | C ≤ ‖g i x‖₊} (g i)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) μ) x
hfx : ¬x ∈ {x | C ≤ ‖f i x‖₊}
⊢ ¬x ∈ {x | C ≤ ‖Exists.choose (_ : AEStronglyMeasurable (f i) μ) x‖₊}
[PROOFSTEP]
rwa [Set.mem_setOf, hx] at hfx
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf : UniformIntegrable f p μ
⊢ UniformIntegrable (fun n => (∑ i in Finset.range n, f i) / ↑n) p μ
[PROOFSTEP]
obtain ⟨hf₁, hf₂, hf₃⟩ := hf
[GOAL]
case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
⊢ UniformIntegrable (fun n => (∑ i in Finset.range n, f i) / ↑n) p μ
[PROOFSTEP]
refine' ⟨fun n => _, fun ε hε => _, _⟩
[GOAL]
case intro.intro.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
⊢ AEStronglyMeasurable ((fun n => (∑ i in Finset.range n, f i) / ↑n) n) μ
[PROOFSTEP]
simp_rw [div_eq_mul_inv]
[GOAL]
case intro.intro.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
⊢ AEStronglyMeasurable ((∑ i in Finset.range n, f i) * (↑n)⁻¹) μ
[PROOFSTEP]
exact
  (Finset.aestronglyMeasurable_sum' _ fun i _ => hf₁ i).mul
    (aestronglyMeasurable_const : AEStronglyMeasurable (fun _ => (↑n : ℝ)⁻¹) μ)
[GOAL]
case intro.intro.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α),
      MeasurableSet s →
        ↑↑μ s ≤ ENNReal.ofReal δ →
          snorm (indicator s ((fun n => (∑ i in Finset.range n, f i) / ↑n) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
obtain ⟨δ, hδ₁, hδ₂⟩ := hf₂ hε
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
⊢ ∃ δ x,
    ∀ (i : ℕ) (s : Set α),
      MeasurableSet s →
        ↑↑μ s ≤ ENNReal.ofReal δ →
          snorm (indicator s ((fun n => (∑ i in Finset.range n, f i) / ↑n) i)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' ⟨δ, hδ₁, fun n s hs hle => _⟩
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (indicator s ((fun n => (∑ i in Finset.range n, f i) / ↑n) n)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp_rw [div_eq_mul_inv, Finset.sum_mul, Set.indicator_finset_sum]
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ snorm (∑ i in Finset.range n, indicator s (f i * (↑n)⁻¹)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' le_trans (snorm_sum_le (fun i _ => ((hf₁ i).mul_const (↑n)⁻¹).indicator hs) hp) _
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ ∑ i in Finset.range n, snorm (indicator s (f i * (↑n)⁻¹)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
have : ∀ i, s.indicator (f i * (n : α → ℝ)⁻¹) = (↑n : ℝ)⁻¹ • s.indicator (f i) :=
  by
  intro i
  rw [mul_comm, (_ : (↑n)⁻¹ * f i = fun ω => (↑n : ℝ)⁻¹ • f i ω)]
  · rw [Set.indicator_const_smul s (↑n : ℝ)⁻¹ (f i)]
    rfl
  · rfl
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
⊢ ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
[PROOFSTEP]
intro i
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
i : ℕ
⊢ indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
[PROOFSTEP]
rw [mul_comm, (_ : (↑n)⁻¹ * f i = fun ω => (↑n : ℝ)⁻¹ • f i ω)]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
i : ℕ
⊢ (indicator s fun ω => (↑n)⁻¹ • f i ω) = (↑n)⁻¹ • indicator s (f i)
[PROOFSTEP]
rw [Set.indicator_const_smul s (↑n : ℝ)⁻¹ (f i)]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
i : ℕ
⊢ (fun x => (↑n)⁻¹ • indicator s (f i) x) = (↑n)⁻¹ • indicator s (f i)
[PROOFSTEP]
rfl
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
i : ℕ
⊢ (↑n)⁻¹ * f i = fun ω => (↑n)⁻¹ • f i ω
[PROOFSTEP]
rfl
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
⊢ ∑ i in Finset.range n, snorm (indicator s (f i * (↑n)⁻¹)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp_rw [this, snorm_const_smul, ← Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat]
[GOAL]
case intro.intro.refine'_2.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp only [hn, zero_mul, zero_le]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ ENNReal.ofReal ε
[PROOFSTEP]
refine' le_trans _ (_ : ↑(↑n : ℝ≥0)⁻¹ * n • ENNReal.ofReal ε ≤ ENNReal.ofReal ε)
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ ↑(↑n)⁻¹ * n • ENNReal.ofReal ε
[PROOFSTEP]
refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
⊢ ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ n • ENNReal.ofReal ε
[PROOFSTEP]
conv_rhs => rw [← Finset.card_range n]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
| n • ENNReal.ofReal ε
[PROOFSTEP]
rw [← Finset.card_range n]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
| n • ENNReal.ofReal ε
[PROOFSTEP]
rw [← Finset.card_range n]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
| n • ENNReal.ofReal ε
[PROOFSTEP]
rw [← Finset.card_range n]
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
⊢ ∑ x in Finset.range n, snorm (indicator s (f x)) p μ ≤ Finset.card (Finset.range n) • ENNReal.ofReal ε
[PROOFSTEP]
exact Finset.sum_le_card_nsmul _ _ _ fun i _ => hδ₂ _ _ hs hle
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * n • ENNReal.ofReal ε ≤ ENNReal.ofReal ε
[PROOFSTEP]
simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬n = 0
⊢ ↑(↑n)⁻¹ * n • ENNReal.ofReal ε ≤ ENNReal.ofReal ε
[PROOFSTEP]
rw [nsmul_eq_mul, ← mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat, ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _),
  one_mul]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬n = 0
⊢ ↑n ≠ 0
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
all_goals simpa only [Ne.def, Nat.cast_eq_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
simpa only [Ne.def, Nat.cast_eq_zero]
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₁ : 0 < δ
hδ₂ :
  ∀ (i : ℕ) (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (indicator s (f i)) p μ ≤ ENNReal.ofReal ε
n : ℕ
s : Set α
hs : MeasurableSet s
hle : ↑↑μ s ≤ ENNReal.ofReal δ
this : ∀ (i : ℕ), indicator s (f i * (↑n)⁻¹) = (↑n)⁻¹ • indicator s (f i)
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
simpa only [Ne.def, Nat.cast_eq_zero]
[GOAL]
case intro.intro.refine'_3
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
hf₃ : ∃ C, ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
⊢ ∃ C, ∀ (i : ℕ), snorm ((fun n => (∑ i in Finset.range n, f i) / ↑n) i) p μ ≤ ↑C
[PROOFSTEP]
obtain ⟨C, hC⟩ := hf₃
[GOAL]
case intro.intro.refine'_3.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
⊢ ∃ C, ∀ (i : ℕ), snorm ((fun n => (∑ i in Finset.range n, f i) / ↑n) i) p μ ≤ ↑C
[PROOFSTEP]
simp_rw [div_eq_mul_inv, Finset.sum_mul]
[GOAL]
case intro.intro.refine'_3.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
⊢ ∃ C, ∀ (i : ℕ), snorm (∑ x in Finset.range i, f x * (↑i)⁻¹) p μ ≤ ↑C
[PROOFSTEP]
refine' ⟨C, fun n => (snorm_sum_le (fun i _ => (hf₁ i).mul_const (↑n)⁻¹) hp).trans _⟩
[GOAL]
case intro.intro.refine'_3.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
⊢ ∑ i in Finset.range n, snorm (fun x => f i x * (↑n)⁻¹) p μ ≤ ↑C
[PROOFSTEP]
have : ∀ i, (fun ω => f i ω * (↑n)⁻¹) = (↑n : ℝ)⁻¹ • fun ω => f i ω :=
  by
  intro i
  ext ω
  simp only [mul_comm, Pi.smul_apply, Algebra.id.smul_eq_mul]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
⊢ ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
[PROOFSTEP]
intro i
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n i : ℕ
⊢ (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
[PROOFSTEP]
ext ω
[GOAL]
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n i : ℕ
ω : α
⊢ f i ω * (↑n)⁻¹ = ((↑n)⁻¹ • fun ω => f i ω) ω
[PROOFSTEP]
simp only [mul_comm, Pi.smul_apply, Algebra.id.smul_eq_mul]
[GOAL]
case intro.intro.refine'_3.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
⊢ ∑ i in Finset.range n, snorm (fun x => f i x * (↑n)⁻¹) p μ ≤ ↑C
[PROOFSTEP]
simp_rw [this, snorm_const_smul, ← Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat]
[GOAL]
case intro.intro.refine'_3.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑C
[PROOFSTEP]
by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0
[GOAL]
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑C
[PROOFSTEP]
simp only [hn, zero_mul, zero_le]
[GOAL]
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑C
[PROOFSTEP]
refine' le_trans _ (_ : ↑(↑n : ℝ≥0)⁻¹ * (n • C : ℝ≥0∞) ≤ C)
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑(↑n)⁻¹ * ↑(n • C)
[PROOFSTEP]
refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑(n • C)
[PROOFSTEP]
conv_rhs =>
  rw [← Finset.card_range n]
    -- Porting note: Originally `exact Finset.sum_le_card_nsmul _ _ _ fun i hi => hC i`
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
| ↑(n • C)
[PROOFSTEP]
rw [← Finset.card_range n]
    -- Porting note: Originally `exact Finset.sum_le_card_nsmul _ _ _ fun i hi => hC i`
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
| ↑(n • C)
[PROOFSTEP]
rw [← Finset.card_range n]
    -- Porting note: Originally `exact Finset.sum_le_card_nsmul _ _ _ fun i hi => hC i`
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
| ↑(n • C)
[PROOFSTEP]
rw [← Finset.card_range n]
  -- Porting note: Originally `exact Finset.sum_le_card_nsmul _ _ _ fun i hi => hC i`
[GOAL]
case neg.refine'_1
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ∑ x in Finset.range n, snorm (fun ω => f x ω) p μ ≤ ↑(Finset.card (Finset.range n) • C)
[PROOFSTEP]
convert Finset.sum_le_card_nsmul _ _ _ fun i _ => hC i
[GOAL]
case h.e'_4
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(Finset.card (Finset.range n) • C) = Finset.card (Finset.range n) • ↑C
[PROOFSTEP]
rw [ENNReal.coe_smul]
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬↑(↑n)⁻¹ = 0
⊢ ↑(↑n)⁻¹ * ↑(n • C) ≤ ↑C
[PROOFSTEP]
simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬n = 0
⊢ ↑(↑n)⁻¹ * ↑(n • C) ≤ ↑C
[PROOFSTEP]
rw [ENNReal.coe_smul, nsmul_eq_mul, ← mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat,
  ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _), one_mul]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬n = 0
⊢ ↑n ≠ 0
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
all_goals simpa only [Ne.def, Nat.cast_eq_zero]
[GOAL]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
simpa only [Ne.def, Nat.cast_eq_zero]
[GOAL]
case neg.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f✝ : ι → α → β
hp : 1 ≤ p
f : ℕ → α → ℝ
hf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ
hf₂ : UnifIntegrable f p μ
C : ℝ≥0
hC : ∀ (i : ℕ), snorm (f i) p μ ≤ ↑C
n : ℕ
this : ∀ (i : ℕ), (fun ω => f i ω * (↑n)⁻¹) = (↑n)⁻¹ • fun ω => f i ω
hn : ¬n = 0
⊢ ↑n ≠ 0
[PROOFSTEP]
simpa only [Ne.def, Nat.cast_eq_zero]