[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
let c : Set ℝ := d '' {s | MeasurableSet s}
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
let γ : ℝ := sSup c
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal <| hμ _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have to_nnreal_ν : ∀ s, ((ν s).toNNReal : ℝ≥0∞) = ν s := fun s => ENNReal.coe_toNNReal <| hν _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have d_split : ∀ s t, MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) :=
  by
  intro s t _hs ht
  dsimp only
  rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _),
    ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add]
  simp only [sub_eq_add_neg, neg_add]
  abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
⊢ ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
[PROOFSTEP]
intro s t _hs ht
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
s t : Set α
_hs : MeasurableSet s
ht : MeasurableSet t
⊢ d s = d (s \ t) + d (s ∩ t)
[PROOFSTEP]
dsimp only
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
s t : Set α
_hs : MeasurableSet s
ht : MeasurableSet t
⊢ ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) =
    ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) +
      (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))
[PROOFSTEP]
rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _),
  ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
s t : Set α
_hs : MeasurableSet s
ht : MeasurableSet t
⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) -
      (↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑ν (s \ t)))) =
    ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) +
      (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))
[PROOFSTEP]
simp only [sub_eq_add_neg, neg_add]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
s t : Set α
_hs : MeasurableSet s
ht : MeasurableSet t
⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) +
      (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t)))) =
    ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t))) +
      (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))
[PROOFSTEP]
abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
s t : Set α
_hs : MeasurableSet s
ht : MeasurableSet t
⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) +
      (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t)))) =
    ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t))) +
      (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))
[PROOFSTEP]
abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have d_Union : ∀ s : ℕ → Set α, Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) :=
  by
  intro s hm
  refine' Tendsto.sub _ _ <;>
    refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm
  exact hμ _
  exact hν _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
⊢ ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
[PROOFSTEP]
intro s hm
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
[PROOFSTEP]
refine' Tendsto.sub _ _
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ Tendsto (fun n => ↑(ENNReal.toNNReal (↑↑μ (s n)))) atTop (𝓝 ↑(ENNReal.toNNReal (↑↑μ (⋃ (n : ℕ), s n))))
[PROOFSTEP]
refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ Tendsto (fun n => ↑(ENNReal.toNNReal (↑↑ν (s n)))) atTop (𝓝 ↑(ENNReal.toNNReal (↑↑ν (⋃ (n : ℕ), s n))))
[PROOFSTEP]
refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ ↑↑μ (⋃ (n : ℕ), s n) ≠ ⊤
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤
[PROOFSTEP]
exact hμ _
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
s : ℕ → Set α
hm : Monotone s
⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤
[PROOFSTEP]
exact hν _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have d_Inter :
  ∀ s : ℕ → Set α,
    (∀ n, MeasurableSet (s n)) → (∀ n m, n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) :=
  by
  intro s hs hm
  refine' Tendsto.sub _ _ <;>
    refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _
  exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
⊢ ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
[PROOFSTEP]
intro s hs hm
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
[PROOFSTEP]
refine' Tendsto.sub _ _
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ Tendsto (fun n => ↑(ENNReal.toNNReal (↑↑μ (s n)))) atTop (𝓝 ↑(ENNReal.toNNReal (↑↑μ (⋂ (n : ℕ), s n))))
[PROOFSTEP]
refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ Tendsto (fun n => ↑(ENNReal.toNNReal (↑↑ν (s n)))) atTop (𝓝 ↑(ENNReal.toNNReal (↑↑ν (⋂ (n : ℕ), s n))))
[PROOFSTEP]
refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _
[GOAL]
case refine'_1.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ ↑↑μ (⋂ (n : ℕ), s n) ≠ ⊤
case refine'_1.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ ∃ i, ↑↑μ (s i) ≠ ⊤
case refine'_2.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ ↑↑ν (⋂ (n : ℕ), s n) ≠ ⊤
case refine'_2.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n
⊢ ∃ i, ↑↑ν (s i) ≠ ⊤
[PROOFSTEP]
exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have bdd_c : BddAbove c := by
  use(μ univ).toNNReal
  rintro r ⟨s, _hs, rfl⟩
  refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _
  rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ]
  exact measure_mono (subset_univ _)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
⊢ BddAbove c
[PROOFSTEP]
use(μ univ).toNNReal
[GOAL]
case h
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
⊢ ↑(ENNReal.toNNReal (↑↑μ univ)) ∈ upperBounds c
[PROOFSTEP]
rintro r ⟨s, _hs, rfl⟩
[GOAL]
case h.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
s : Set α
_hs : s ∈ {s | MeasurableSet s}
⊢ d s ≤ ↑(ENNReal.toNNReal (↑↑μ univ))
[PROOFSTEP]
refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _
[GOAL]
case h.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
s : Set α
_hs : s ∈ {s | MeasurableSet s}
⊢ ↑(ENNReal.toNNReal (↑↑μ s)) ≤ ↑(ENNReal.toNNReal (↑↑μ univ))
[PROOFSTEP]
rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ]
[GOAL]
case h.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
s : Set α
_hs : s ∈ {s | MeasurableSet s}
⊢ ↑↑μ s ≤ ↑↑μ univ
[PROOFSTEP]
exact measure_mono (subset_univ _)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have c_nonempty : c.Nonempty := Nonempty.image _ ⟨_, MeasurableSet.empty⟩
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have d_le_γ : ∀ s, MeasurableSet s → d s ≤ γ := fun s hs => le_csSup bdd_c ⟨s, hs, rfl⟩
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have : ∀ n : ℕ, ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s :=
  by
  intro n
  have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)
  rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩
  exact ⟨s, hs, hlt⟩
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
⊢ ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
[PROOFSTEP]
intro n
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
n : ℕ
⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
[PROOFSTEP]
have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
n : ℕ
this : γ - (1 / 2) ^ n < γ
⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
[PROOFSTEP]
rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩
[GOAL]
case intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
n : ℕ
this : γ - (1 / 2) ^ n < γ
s : Set α
hs : s ∈ {s | MeasurableSet s}
hlt : γ - (1 / 2) ^ n < d s
⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
[PROOFSTEP]
exact ⟨s, hs, hlt⟩
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
rcases Classical.axiom_of_choice this with ⟨e, he⟩
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
change ℕ → Set α at e
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
let f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have hf : ∀ n m, MeasurableSet (f n m) := by
  intro n m
  simp only [Finset.inf_eq_iInf]
  exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
⊢ ∀ (n m : ℕ), MeasurableSet (f n m)
[PROOFSTEP]
intro n m
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
n m : ℕ
⊢ MeasurableSet (f n m)
[PROOFSTEP]
simp only [Finset.inf_eq_iInf]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
n m : ℕ
⊢ MeasurableSet (⨅ (a : ℕ) (_ : a ∈ Finset.Ico n (m + 1)), e a)
[PROOFSTEP]
exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have f_subset_f : ∀ {a b c d}, a ≤ b → c ≤ d → f a d ⊆ f b c :=
  by
  intro a b c d hab hcd
  simp_rw [Finset.inf_eq_iInf]
  exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
⊢ ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
[PROOFSTEP]
intro a b c d hab hcd
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c✝ : Set ℝ := d✝ '' {s | MeasurableSet s}
γ : ℝ := sSup c✝
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c✝
c_nonempty : Set.Nonempty c✝
d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
a b c d : ℕ
hab : a ≤ b
hcd : c ≤ d
⊢ f a d ⊆ f b c
[PROOFSTEP]
simp_rw [Finset.inf_eq_iInf]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c✝ : Set ℝ := d✝ '' {s | MeasurableSet s}
γ : ℝ := sSup c✝
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c✝
c_nonempty : Set.Nonempty c✝
d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
a b c d : ℕ
hab : a ≤ b
hcd : c ≤ d
⊢ ⨅ (a_1 : ℕ) (_ : a_1 ∈ Finset.Ico a (d + 1)), e a_1 ⊆ ⨅ (a : ℕ) (_ : a ∈ Finset.Ico b (c + 1)), e a
[PROOFSTEP]
exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have f_succ : ∀ n m, n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) :=
  by
  intro n m hnm
  have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)
  simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]
  rfl
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
⊢ ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
[PROOFSTEP]
intro n m hnm
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
⊢ f n (m + 1) = f n m ∩ e (m + 1)
[PROOFSTEP]
have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
this : n ≤ m + 1
⊢ f n (m + 1) = f n m ∩ e (m + 1)
[PROOFSTEP]
simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
this : n ≤ m + 1
⊢ e (m + 1) ⊓ Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) ∩ Finset.inf (Finset.Ico n (m + 1)) e
[PROOFSTEP]
rfl
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have le_d_f : ∀ n m, m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) :=
  by
  intro n m h
  refine' Nat.le_induction _ _ n h
  · have := he₂ m
    simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]
    linarith
  · intro n (hmn : m ≤ n) ih
    have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by
      calc
        γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=
          by
          refine' add_le_add_left (add_le_add_left _ _) γ
          simp only [pow_add, pow_one, le_sub_iff_add_le]
          linarith
        _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel
        _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he₂ _) ih)
        _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by
          rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]
        _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) :=
          by
          rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]
          abel
          exact (he₁ _).union (hf _ _)
          exact he₁ _
        _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <| (he₁ _).union (hf _ _)) _
    exact (add_le_add_iff_left γ).1 this
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
⊢ ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
[PROOFSTEP]
intro n m h
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n m : ℕ
h : m ≤ n
⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
[PROOFSTEP]
refine' Nat.le_induction _ _ n h
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n m : ℕ
h : m ≤ n
⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)
[PROOFSTEP]
have := he₂ m
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n m : ℕ
h : m ≤ n
this : γ - (1 / 2) ^ m < d (e m)
⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)
[PROOFSTEP]
simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n m : ℕ
h : m ≤ n
this : γ - (1 / 2) ^ m < d (e m)
⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) - ↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) -
        2 * (1 / 2) ^ m +
      (1 / 2) ^ m ≤
    ↑(ENNReal.toNNReal (↑↑μ (e m))) - ↑(ENNReal.toNNReal (↑↑ν (e m)))
[PROOFSTEP]
linarith
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n m : ℕ
h : m ≤ n
⊢ ∀ (n : ℕ),
    m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))
[PROOFSTEP]
intro n (hmn : m ≤ n) ih
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))
[PROOFSTEP]
have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by
  calc
    γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=
      by
      refine' add_le_add_left (add_le_add_left _ _) γ
      simp only [pow_add, pow_one, le_sub_iff_add_le]
      linarith
    _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel
    _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he₂ _) ih)
    _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by
      rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]
    _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) :=
      by
      rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]
      abel
      exact (he₁ _).union (hf _ _)
      exact he₁ _
    _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <| (he₁ _).union (hf _ _)) _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))
[PROOFSTEP]
calc
  γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=
    by
    refine' add_le_add_left (add_le_add_left _ _) γ
    simp only [pow_add, pow_one, le_sub_iff_add_le]
    linarith
  _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel
  _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he₂ _) ih)
  _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by
    rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]
  _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) :=
    by
    rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]
    abel
    exact (he₁ _).union (hf _ _)
    exact he₁ _
  _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <| (he₁ _).union (hf _ _)) _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1)))
[PROOFSTEP]
refine' add_le_add_left (add_le_add_left _ _) γ
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1)
[PROOFSTEP]
simp only [pow_add, pow_one, le_sub_iff_add_le]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) ≤ (1 / 2) ^ n
[PROOFSTEP]
linarith
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) =
    γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n)
[PROOFSTEP]
simp only [sub_eq_add_neg]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
      (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
          -(2 * (1 / 2) ^ m) +
        ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =
    sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
        -(1 / 2) ^ (n + 1) +
      (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
          -(2 * (1 / 2) ^ m) +
        (1 / 2) ^ n)
[PROOFSTEP]
abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
      (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
          -(2 * (1 / 2) ^ m) +
        ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =
    sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
        -(1 / 2) ^ (n + 1) +
      (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +
          -(2 * (1 / 2) ^ m) +
        (1 / 2) ^ n)
[PROOFSTEP]
abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ d (e (n + 1)) + d (f m n) ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1))
[PROOFSTEP]
rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (e (n + 1) ∪ f m n) + d (f m (n + 1))
[PROOFSTEP]
rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (f m n \ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1) ∪ f m n)
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1))
[PROOFSTEP]
abel
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (f m n \ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1) ∪ f m n)
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1))
[PROOFSTEP]
abel
[GOAL]
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1) ∪ f m n)
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1))
[PROOFSTEP]
exact (he₁ _).union (hf _ _)
[GOAL]
case a
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1))
[PROOFSTEP]
exact he₁ _
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
this : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))
⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))
[PROOFSTEP]
exact (add_le_add_iff_left γ).1 this
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
let s := ⋃ m, ⋂ n, f m n
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have γ_le_d_s : γ ≤ d s :=
  by
  have hγ : Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) :=
    by
    suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero]
    exact
      tendsto_const_nhds.sub <|
        tendsto_const_nhds.mul <|
          tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)
  have hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) :=
    by
    refine' d_Union _ _
    exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl
  refine' le_of_tendsto_of_tendsto' hγ hd fun m => _
  have : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) :=
    by
    refine' d_Inter _ _ _
    · intro n
      exact hf _ _
    · intro n m hnm
      exact f_subset_f le_rfl hnm
  refine' ge_of_tendsto this (eventually_atTop.2 ⟨m, fun n hmn => _⟩)
  change γ - 2 * (1 / 2) ^ m ≤ d (f m n)
  refine' le_trans _ (le_d_f _ _ hmn)
  exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <| half_pos <| zero_lt_one) _)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
⊢ γ ≤ d s
[PROOFSTEP]
have hγ : Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) :=
  by
  suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero]
  exact
    tendsto_const_nhds.sub <|
      tendsto_const_nhds.mul <|
        tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
[PROOFSTEP]
suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
this : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))
⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
[PROOFSTEP]
simpa only [mul_zero, tsub_zero]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))
[PROOFSTEP]
exact
  tendsto_const_nhds.sub <|
    tendsto_const_nhds.mul <|
      tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
⊢ γ ≤ d s
[PROOFSTEP]
have hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) :=
  by
  refine' d_Union _ _
  exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
⊢ Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
[PROOFSTEP]
refine' d_Union _ _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
⊢ Monotone fun m => ⋂ (n : ℕ), f m n
[PROOFSTEP]
exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
⊢ γ ≤ d s
[PROOFSTEP]
refine' le_of_tendsto_of_tendsto' hγ hd fun m => _
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ (n : ℕ), f m n)
[PROOFSTEP]
have : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) :=
  by
  refine' d_Inter _ _ _
  · intro n
    exact hf _ _
  · intro n m hnm
    exact f_subset_f le_rfl hnm
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
⊢ Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))
[PROOFSTEP]
refine' d_Inter _ _ _
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
⊢ ∀ (n : ℕ), MeasurableSet (f m n)
[PROOFSTEP]
intro n
[GOAL]
case refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m n : ℕ
⊢ MeasurableSet (f m n)
[PROOFSTEP]
exact hf _ _
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n
[PROOFSTEP]
intro n m hnm
[GOAL]
case refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m✝ n m : ℕ
hnm : n ≤ m
⊢ f m✝ m ⊆ f m✝ n
[PROOFSTEP]
exact f_subset_f le_rfl hnm
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))
⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ (n : ℕ), f m n)
[PROOFSTEP]
refine' ge_of_tendsto this (eventually_atTop.2 ⟨m, fun n hmn => _⟩)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))
n : ℕ
hmn : n ≥ m
⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)
[PROOFSTEP]
change γ - 2 * (1 / 2) ^ m ≤ d (f m n)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))
n : ℕ
hmn : n ≥ m
⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)
[PROOFSTEP]
refine' le_trans _ (le_d_f _ _ hmn)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))
m : ℕ
this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))
n : ℕ
hmn : n ≥ m
⊢ γ - 2 * (1 / 2) ^ m ≤ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n
[PROOFSTEP]
exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <| half_pos <| zero_lt_one) _)
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have hs : MeasurableSet s := MeasurableSet.iUnion fun n => MeasurableSet.iInter fun m => hf _ _
[GOAL]
case intro
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
⊢ ∃ s,
    MeasurableSet s ∧
      (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
refine' ⟨s, hs, _, _⟩
[GOAL]
case intro.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t
[PROOFSTEP]
intro t ht hts
[GOAL]
case intro.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
⊢ ↑↑ν t ≤ ↑↑μ t
[PROOFSTEP]
have : 0 ≤ d t :=
  (add_le_add_iff_left γ).1 <|
    calc
      γ + 0 ≤ d s := by rw [add_zero]; exact γ_le_d_s
      _ = d (s \ t) + d t := by rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]
      _ ≤ γ + d t := add_le_add (d_le_γ _ (hs.diff ht)) le_rfl
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
⊢ γ + 0 ≤ d s
[PROOFSTEP]
rw [add_zero]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
⊢ γ ≤ d s
[PROOFSTEP]
exact γ_le_d_s
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
⊢ d s = d (s \ t) + d t
[PROOFSTEP]
rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]
[GOAL]
case intro.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
this : 0 ≤ d t
⊢ ↑↑ν t ≤ ↑↑μ t
[PROOFSTEP]
rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe]
[GOAL]
case intro.refine'_1
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ s
this : 0 ≤ d t
⊢ ↑(ENNReal.toNNReal (↑↑ν t)) ≤ ↑(ENNReal.toNNReal (↑↑μ t))
[PROOFSTEP]
simpa only [le_sub_iff_add_le, zero_add] using this
[GOAL]
case intro.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
intro t ht hts
[GOAL]
case intro.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
have : d t ≤ 0 :=
  (add_le_add_iff_left γ).1 <|
    calc
      γ + d t ≤ d s + d t := add_le_add γ_le_d_s le_rfl
      _ = d (s ∪ t) := by
        rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,
          (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]
      _ ≤ γ + 0 := by rw [add_zero]; exact d_le_γ _ (hs.union ht)
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ d s + d t = d (s ∪ t)
[PROOFSTEP]
rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,
  (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ d (s ∪ t) ≤ γ + 0
[PROOFSTEP]
rw [add_zero]
[GOAL]
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ d (s ∪ t) ≤ γ
[PROOFSTEP]
exact d_le_γ _ (hs.union ht)
[GOAL]
case intro.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
this : d t ≤ 0
⊢ ↑↑μ t ≤ ↑↑ν t
[PROOFSTEP]
rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe]
[GOAL]
case intro.refine'_2
α : Type u_1
inst✝² : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤
hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s
d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : Set.Nonempty c
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
this : d t ≤ 0
⊢ ↑(ENNReal.toNNReal (↑↑μ t)) ≤ ↑(ENNReal.toNNReal (↑↑ν t))
[PROOFSTEP]
simpa only [sub_le_iff_le_add, zero_add] using this