[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
⊢ ∃ T,
    Set.Countable T ∧
      (∀ (u : Set X), u ∈ T → IsOpen u) ∧
        s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
[PROOFSTEP]
have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ :=
  by
  rintro ⟨x, hx⟩
  rcases hf x hx with ⟨t, ht, h't⟩
  rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
  refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
⊢ ∀ (x : ↑s), ∃ u, IsOpen u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s)
[PROOFSTEP]
rintro ⟨x, hx⟩
[GOAL]
case mk
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
x : X
hx : x ∈ s
⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s)
[PROOFSTEP]
rcases hf x hx with ⟨t, ht, h't⟩
[GOAL]
case mk.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
x : X
hx : x ∈ s
t : Set X
ht : t ∈ 𝓝[s] x
h't : IntegrableOn f t
⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s)
[PROOFSTEP]
rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
[GOAL]
case mk.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
x : X
hx : x ∈ s
t : Set X
ht : t ∈ 𝓝[s] x
h't : IntegrableOn f t
u : Set X
u_open : IsOpen u
x_mem : x ∈ u
u_sub : u ∩ s ⊆ t
⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s)
[PROOFSTEP]
refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
this : ∀ (x : ↑s), ∃ u, IsOpen u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s)
⊢ ∃ T,
    Set.Countable T ∧
      (∀ (u : Set X), u ∈ T → IsOpen u) ∧
        s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
[PROOFSTEP]
choose u u_open xu hu using this
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
⊢ ∃ T,
    Set.Countable T ∧
      (∀ (u : Set X), u ∈ T → IsOpen u) ∧
        s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
[PROOFSTEP]
obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s ⊆ ⋃ i ∈ T, u i :=
  by
  have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
  obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open
  exact ⟨T, hT_count, by rwa [hT_un]⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
[PROOFSTEP]
have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
this : s ⊆ ⋃ (x : ↑s), u x
⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
[PROOFSTEP]
obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
this : s ⊆ ⋃ (x : ↑s), u x
T : Set ↑s
hT_count : Set.Countable T
hT_un : ⋃ (i : ↑s) (_ : i ∈ T), u i = ⋃ (i : ↑s), u i
⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
[PROOFSTEP]
exact ⟨T, hT_count, by rwa [hT_un]⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
this : s ⊆ ⋃ (x : ↑s), u x
T : Set ↑s
hT_count : Set.Countable T
hT_un : ⋃ (i : ↑s) (_ : i ∈ T), u i = ⋃ (i : ↑s), u i
⊢ s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
[PROOFSTEP]
rwa [hT_un]
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
⊢ ∃ T,
    Set.Countable T ∧
      (∀ (u : Set X), u ∈ T → IsOpen u) ∧
        s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
[PROOFSTEP]
refine' ⟨u '' T, T_count.image _, _, by rwa [biUnion_image], _⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
⊢ s ⊆ ⋃ (u_1 : Set X) (_ : u_1 ∈ u '' T), u_1
[PROOFSTEP]
rwa [biUnion_image]
[GOAL]
case intro.intro.refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
⊢ ∀ (u_1 : Set X), u_1 ∈ u '' T → IsOpen u_1
[PROOFSTEP]
rintro v ⟨w, -, rfl⟩
[GOAL]
case intro.intro.refine'_1.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
w : ↑s
⊢ IsOpen (u w)
[PROOFSTEP]
exact u_open _
[GOAL]
case intro.intro.refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
⊢ ∀ (u_1 : Set X), u_1 ∈ u '' T → IntegrableOn f (u_1 ∩ s)
[PROOFSTEP]
rintro v ⟨w, -, rfl⟩
[GOAL]
case intro.intro.refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s)
T : Set ↑s
T_count : Set.Countable T
hT : s ⊆ ⋃ (i : ↑s) (_ : i ∈ T), u i
w : ↑s
⊢ IntegrableOn f (u w ∩ s)
[PROOFSTEP]
exact hu _
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
[GOAL]
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
let T' : Set (Set X) := insert ∅ T
[GOAL]
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
have T'_count : T'.Countable := Countable.insert ∅ T_count
[GOAL]
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
⊢ Set.Nonempty T'
[PROOFSTEP]
simp only [insert_nonempty]
[GOAL]
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
[GOAL]
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ (n : ℕ), u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
refine' ⟨u, _, _, _⟩
[GOAL]
case intro.intro.intro.intro.intro.refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
⊢ ∀ (n : ℕ), IsOpen (u n)
[PROOFSTEP]
intro n
[GOAL]
case intro.intro.intro.intro.intro.refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ IsOpen (u n)
[PROOFSTEP]
have : u n ∈ T' := by rw [hu]; exact mem_range_self n
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ T'
[PROOFSTEP]
rw [hu]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ range u
[PROOFSTEP]
exact mem_range_self n
[GOAL]
case intro.intro.intro.intro.intro.refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
⊢ IsOpen (u n)
[PROOFSTEP]
rcases mem_insert_iff.1 this with h | h
[GOAL]
case intro.intro.intro.intro.intro.refine'_1.inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen (u n)
[PROOFSTEP]
rw [h]
[GOAL]
case intro.intro.intro.intro.intro.refine'_1.inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen ∅
[PROOFSTEP]
exact isOpen_empty
[GOAL]
case intro.intro.intro.intro.intro.refine'_1.inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n ∈ T
⊢ IsOpen (u n)
[PROOFSTEP]
exact T_open _ h
[GOAL]
case intro.intro.intro.intro.intro.refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
⊢ s ⊆ ⋃ (n : ℕ), u n
[PROOFSTEP]
intro x hx
[GOAL]
case intro.intro.intro.intro.intro.refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
⊢ x ∈ ⋃ (n : ℕ), u n
[PROOFSTEP]
obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
⊢ ∃ v, v ∈ T ∧ x ∈ v
[PROOFSTEP]
simpa only [mem_iUnion, exists_prop] using sT hx
[GOAL]
case intro.intro.intro.intro.intro.refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ x ∈ ⋃ (n : ℕ), u n
[PROOFSTEP]
have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ range u
[PROOFSTEP]
rw [← hu]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ T'
[PROOFSTEP]
exact subset_insert ∅ T hv
[GOAL]
case intro.intro.intro.intro.intro.refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
this : v ∈ range u
⊢ x ∈ ⋃ (n : ℕ), u n
[PROOFSTEP]
obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
this : v ∈ range u
⊢ ∃ n, u n = v
[PROOFSTEP]
simpa only [mem_range] using this
[GOAL]
case intro.intro.intro.intro.intro.refine'_2.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
n : ℕ
hv : u n ∈ T
h'v : x ∈ u n
this : u n ∈ range u
⊢ x ∈ ⋃ (n : ℕ), u n
[PROOFSTEP]
exact mem_iUnion_of_mem _ h'v
[GOAL]
case intro.intro.intro.intro.intro.refine'_3
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
⊢ ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
[PROOFSTEP]
intro n
[GOAL]
case intro.intro.intro.intro.intro.refine'_3
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ IntegrableOn f (u n ∩ s)
[PROOFSTEP]
have : u n ∈ T' := by rw [hu]; exact mem_range_self n
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ T'
[PROOFSTEP]
rw [hu]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ range u
[PROOFSTEP]
exact mem_range_self n
[GOAL]
case intro.intro.intro.intro.intro.refine'_3
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
⊢ IntegrableOn f (u n ∩ s)
[PROOFSTEP]
rcases mem_insert_iff.1 this with h | h
[GOAL]
case intro.intro.intro.intro.intro.refine'_3.inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IntegrableOn f (u n ∩ s)
[PROOFSTEP]
simp only [h, empty_inter, integrableOn_empty]
[GOAL]
case intro.intro.intro.intro.intro.refine'_3.inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
T : Set (Set X)
T_count : Set.Countable T
T_open : ∀ (u : Set X), u ∈ T → IsOpen u
sT : s ⊆ ⋃ (u : Set X) (_ : u ∈ T), u
hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s)
T' : Set (Set X) := insert ∅ T
T'_count : Set.Countable T'
T'_ne : Set.Nonempty T'
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n ∈ T
⊢ IntegrableOn f (u n ∩ s)
[PROOFSTEP]
exact hT _ h
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
⊢ AEStronglyMeasurable f (Measure.restrict μ s)
[PROOFSTEP]
rcases hf.exists_nat_integrableOn with ⟨u, -, su, hu⟩
[GOAL]
case intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ℕ → Set X
su : s ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
⊢ AEStronglyMeasurable f (Measure.restrict μ s)
[PROOFSTEP]
have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right_iff_subset.mpr su).symm
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ℕ → Set X
su : s ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
⊢ s = ⋃ (n : ℕ), u n ∩ s
[PROOFSTEP]
rw [← iUnion_inter]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ℕ → Set X
su : s ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
⊢ s = (⋃ (i : ℕ), u i) ∩ s
[PROOFSTEP]
exact (inter_eq_right_iff_subset.mpr su).symm
[GOAL]
case intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ℕ → Set X
su : s ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
this : s = ⋃ (n : ℕ), u n ∩ s
⊢ AEStronglyMeasurable f (Measure.restrict μ s)
[PROOFSTEP]
rw [this, aestronglyMeasurable_iUnion_iff]
[GOAL]
case intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s
u : ℕ → Set X
su : s ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s)
this : s = ⋃ (n : ℕ), u n ∩ s
⊢ ∀ (i : ℕ), AEStronglyMeasurable f (Measure.restrict μ (u i ∩ s))
[PROOFSTEP]
exact fun i : ℕ => (hu i).aestronglyMeasurable
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hs : IsClosed s ∨ IsOpen s
⊢ LocallyIntegrableOn f s ↔ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
[PROOFSTEP]
refine' ⟨fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => _⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hs : IsClosed s ∨ IsOpen s
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
cases hs with
| inl hs =>
  exact
    let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
    ⟨_, inter_mem_nhdsWithin s h2K,
      hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
| inr hs =>
  obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
  refine' ⟨K, _, hf K h3K hK⟩
  simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hs : IsClosed s ∨ IsOpen s
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
cases hs with
| inl hs =>
  exact
    let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
    ⟨_, inter_mem_nhdsWithin s h2K,
      hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
| inr hs =>
  obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
  refine' ⟨K, _, hf K h3K hK⟩
  simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
[GOAL]
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsClosed s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]

| inl hs =>
  exact
    let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
    ⟨_, inter_mem_nhdsWithin s h2K,
      hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
[GOAL]
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsClosed s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
exact
  let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
  ⟨_, inter_mem_nhdsWithin s h2K,
    hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsOpen s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]

| inr hs =>
  obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
  refine' ⟨K, _, hf K h3K hK⟩
  simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsOpen s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
[GOAL]
case inr.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsOpen s
K : Set X
hK : IsCompact K
h2K : x ∈ interior K
h3K : K ⊆ s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
refine' ⟨K, _, hf K h3K hK⟩
[GOAL]
case inr.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝¹ : LocallyCompactSpace X
inst✝ : T2Space X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
x : X
hx : x ∈ s
hs : IsOpen s
K : Set X
hK : IsCompact K
h2K : x ∈ interior K
h3K : K ⊆ s
⊢ K ∈ 𝓝[s] x
[PROOFSTEP]
simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
⊢ LocallyIntegrableOn f univ ↔ LocallyIntegrable f
[PROOFSTEP]
simp only [LocallyIntegrableOn, nhdsWithin_univ, mem_univ, true_imp_iff]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
⊢ (∀ (x : X), IntegrableAtFilter f (𝓝 x)) ↔ LocallyIntegrable f
[PROOFSTEP]
rfl
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f
⊢ LocallyIntegrableOn f s
[PROOFSTEP]
intro x _
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f
x : X
a✝ : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
obtain ⟨t, ht_mem, ht_int⟩ := hf x
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f
x : X
a✝ : x ∈ s
t : Set X
ht_mem : t ∈ 𝓝 x
ht_int : IntegrableOn f t
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
obtain ⟨u, hu_sub, hu_o, hu_mem⟩ := mem_nhds_iff.mp ht_mem
[GOAL]
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f
x : X
a✝ : x ∈ s
t : Set X
ht_mem : t ∈ 𝓝 x
ht_int : IntegrableOn f t
u : Set X
hu_sub : u ⊆ t
hu_o : IsOpen u
hu_mem : x ∈ u
⊢ IntegrableAtFilter f (𝓝[s] x)
[PROOFSTEP]
refine' ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _⟩
[GOAL]
case intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f
x : X
a✝ : x ∈ s
t : Set X
ht_mem : t ∈ 𝓝 x
ht_int : IntegrableOn f t
u : Set X
hu_sub : u ⊆ t
hu_o : IsOpen u
hu_mem : x ∈ u
⊢ IntegrableOn f (s ∩ u)
[PROOFSTEP]
simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using ht_int.mono_set hu_sub
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
⊢ LocallyIntegrableOn f s ↔ LocallyIntegrable f
[PROOFSTEP]
refine' ⟨fun hf x => _, locallyIntegrableOn_of_locallyIntegrable_restrict⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
by_cases h : x ∈ s
[GOAL]
case pos
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : x ∈ s
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
obtain ⟨t, ht_nhds, ht_int⟩ := hf x h
[GOAL]
case pos.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhdsWithin.mp ht_nhds
[GOAL]
case pos.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t
u : Set X
hu_o : IsOpen u
hu_x : x ∈ u
hu_sub : u ∩ s ⊆ t
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
refine' ⟨u, hu_o.mem_nhds hu_x, _⟩
[GOAL]
case pos.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t
u : Set X
hu_o : IsOpen u
hu_x : x ∈ u
hu_sub : u ∩ s ⊆ t
⊢ IntegrableOn f u
[PROOFSTEP]
rw [IntegrableOn, restrict_restrict hu_o.measurableSet]
[GOAL]
case pos.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : IntegrableOn f t
u : Set X
hu_o : IsOpen u
hu_x : x ∈ u
hu_sub : u ∩ s ⊆ t
⊢ Integrable f
[PROOFSTEP]
exact ht_int.mono_set hu_sub
[GOAL]
case neg
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s
x : X
h : ¬x ∈ s
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
rw [← isOpen_compl_iff] at hs
[GOAL]
case neg
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen sᶜ
hf : LocallyIntegrableOn f s
x : X
h : ¬x ∈ s
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
refine' ⟨sᶜ, hs.mem_nhds h, _⟩
[GOAL]
case neg
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen sᶜ
hf : LocallyIntegrableOn f s
x : X
h : ¬x ∈ s
⊢ IntegrableOn f sᶜ
[PROOFSTEP]
rw [IntegrableOn, restrict_restrict, inter_comm, inter_compl_self, ← IntegrableOn]
[GOAL]
case neg
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen sᶜ
hf : LocallyIntegrableOn f s
x : X
h : ¬x ∈ s
⊢ IntegrableOn f ∅
case neg
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen sᶜ
hf : LocallyIntegrableOn f s
x : X
h : ¬x ∈ s
⊢ MeasurableSet sᶜ
[PROOFSTEP]
exacts [integrableOn_empty, hs.measurableSet]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∃ u, IsOpen u ∧ k ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
refine' IsCompact.induction_on hk _ _ _ _
[GOAL]
case refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∃ u, IsOpen u ∧ ∅ ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
refine' ⟨∅, isOpen_empty, Subset.rfl, integrableOn_empty⟩
[GOAL]
case refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
rintro s t hst ⟨u, u_open, tu, hu⟩
[GOAL]
case refine'_2.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
s t : Set X
hst : s ⊆ t
u : Set X
u_open : IsOpen u
tu : t ⊆ u
hu : IntegrableOn f u
⊢ ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
exact ⟨u, u_open, hst.trans tu, hu⟩
[GOAL]
case refine'_3
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ ⦃s t : Set X⦄,
    (∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u) →
      (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
rintro s t ⟨u, u_open, su, hu⟩ ⟨v, v_open, tv, hv⟩
[GOAL]
case refine'_3.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
s t u : Set X
u_open : IsOpen u
su : s ⊆ u
hu : IntegrableOn f u
v : Set X
v_open : IsOpen v
tv : t ⊆ v
hv : IntegrableOn f v
⊢ ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
exact ⟨u ∪ v, u_open.union v_open, union_subset_union su tv, hu.union hv⟩
[GOAL]
case refine'_4
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
⊢ ∀ (x : X), x ∈ k → ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
intro x _
[GOAL]
case refine'_4
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
rcases hf x with ⟨u, ux, hu⟩
[GOAL]
case refine'_4.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
u : Set X
ux : u ∈ 𝓝 x
hu : IntegrableOn f u
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
rcases mem_nhds_iff.1 ux with ⟨v, vu, v_open, xv⟩
[GOAL]
case refine'_4.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
hf : LocallyIntegrable f
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
u : Set X
ux : u ∈ 𝓝 x
hu : IntegrableOn f u
v : Set X
vu : v ⊆ u
v_open : IsOpen v
xv : x ∈ v
⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u
[PROOFSTEP]
exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, Subset.rfl, hu.mono_set vu⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrable f
⊢ AEStronglyMeasurable f μ
[PROOFSTEP]
simpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrable f
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ ⋃ (n : ℕ), u n = univ ∧ ∀ (n : ℕ), IntegrableOn f (u n)
[PROOFSTEP]
rcases(hf.locallyIntegrableOn univ).exists_nat_integrableOn with ⟨u, u_open, u_union, hu⟩
[GOAL]
case intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrable f
u : ℕ → Set X
u_open : ∀ (n : ℕ), IsOpen (u n)
u_union : univ ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ univ)
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ ⋃ (n : ℕ), u n = univ ∧ ∀ (n : ℕ), IntegrableOn f (u n)
[PROOFSTEP]
refine' ⟨u, u_open, eq_univ_of_univ_subset u_union, fun n ↦ _⟩
[GOAL]
case intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrable f
u : ℕ → Set X
u_open : ∀ (n : ℕ), IsOpen (u n)
u_union : univ ⊆ ⋃ (n : ℕ), u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ univ)
n : ℕ
⊢ IntegrableOn f (u n)
[PROOFSTEP]
simpa only [inter_univ] using hu n
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : IsLocallyFiniteMeasure μ
c : E
⊢ LocallyIntegrable fun x => c
[PROOFSTEP]
intro x
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : IsLocallyFiniteMeasure μ
c : E
x : X
⊢ IntegrableAtFilter (fun x => c) (𝓝 x)
[PROOFSTEP]
rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : IsLocallyFiniteMeasure μ
c : E
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ IntegrableAtFilter (fun x => c) (𝓝 x)
[PROOFSTEP]
refine' ⟨U, hU, _⟩
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : MeasurableSpace Y
inst✝² : TopologicalSpace Y
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : IsLocallyFiniteMeasure μ
c : E
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ IntegrableOn (fun x => c) U
[PROOFSTEP]
simp only [h'U, integrableOn_const, or_true_iff]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
s : Set X
hs : MeasurableSet s
⊢ LocallyIntegrable (Set.indicator s f)
[PROOFSTEP]
intro x
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
s : Set X
hs : MeasurableSet s
x : X
⊢ IntegrableAtFilter (Set.indicator s f) (𝓝 x)
[PROOFSTEP]
rcases hf x with ⟨U, hU, h'U⟩
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : TopologicalSpace Y
inst✝ : NormedAddCommGroup E
f : X → E
μ : Measure X
s✝ : Set X
hf : LocallyIntegrable f
s : Set X
hs : MeasurableSet s
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : IntegrableOn f U
⊢ IntegrableAtFilter (Set.indicator s f) (𝓝 x)
[PROOFSTEP]
exact ⟨U, hU, h'U.indicator hs⟩
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
⊢ LocallyIntegrable f ↔ LocallyIntegrable (f ∘ ↑e)
[PROOFSTEP]
refine' ⟨fun h x => _, fun h x => _⟩
[GOAL]
case refine'_1
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable f
x : X
⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)
[PROOFSTEP]
rcases h (e x) with ⟨U, hU, h'U⟩
[GOAL]
case refine'_1.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable f
x : X
U : Set Y
hU : U ∈ 𝓝 (↑e x)
h'U : IntegrableOn f U
⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)
[PROOFSTEP]
refine' ⟨e ⁻¹' U, e.continuous.continuousAt.preimage_mem_nhds hU, _⟩
[GOAL]
case refine'_1.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable f
x : X
U : Set Y
hU : U ∈ 𝓝 (↑e x)
h'U : IntegrableOn f U
⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' U)
[PROOFSTEP]
exact (integrableOn_map_equiv e.toMeasurableEquiv).1 h'U
[GOAL]
case refine'_2
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
rcases h (e.symm x) with ⟨U, hU, h'U⟩
[GOAL]
case refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)
h'U : IntegrableOn (f ∘ ↑e) U
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
refine' ⟨e.symm ⁻¹' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, _⟩
[GOAL]
case refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)
h'U : IntegrableOn (f ∘ ↑e) U
⊢ IntegrableOn f (↑(Homeomorph.symm e) ⁻¹' U)
[PROOFSTEP]
apply (integrableOn_map_equiv e.toMeasurableEquiv).2
[GOAL]
case refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)
h'U : IntegrableOn (f ∘ ↑e) U
⊢ IntegrableOn (f ∘ ↑(Homeomorph.toMeasurableEquiv e))
    (↑(Homeomorph.toMeasurableEquiv e) ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U))
[PROOFSTEP]
simp only [Homeomorph.toMeasurableEquiv_coe]
[GOAL]
case refine'_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)
h'U : IntegrableOn (f ∘ ↑e) U
⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U))
[PROOFSTEP]
convert h'U
[GOAL]
case h.e'_6
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)
h'U : IntegrableOn (f ∘ ↑e) U
⊢ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) = U
[PROOFSTEP]
ext x
[GOAL]
case h.e'_6.h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
f✝ : X → E
μ✝ : Measure X
s : Set X
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ↑e)
x✝ : Y
U : Set X
hU : U ∈ 𝓝 (↑(Homeomorph.symm e) x✝)
h'U : IntegrableOn (f ∘ ↑e) U
x : X
⊢ x ∈ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) ↔ x ∈ U
[PROOFSTEP]
simp only [mem_preimage, Homeomorph.symm_apply_apply]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
⊢ Integrable fun x => g x • f x
[PROOFSTEP]
let K := tsupport g
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
⊢ Integrable fun x => g x • f x
[PROOFSTEP]
have hK : IsCompact K := h'g
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ Integrable fun x => g x • f x
[PROOFSTEP]
have : K.indicator (fun x ↦ g x • f x) = (fun x ↦ g x • f x) :=
  by
  apply indicator_eq_self.2
  apply support_subset_iff'.2
  intros x hx
  simp [image_eq_zero_of_nmem_tsupport hx]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (Set.indicator K fun x => g x • f x) = fun x => g x • f x
[PROOFSTEP]
apply indicator_eq_self.2
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (support fun x => g x • f x) ⊆ K
[PROOFSTEP]
apply support_subset_iff'.2
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ ∀ (x : X), ¬x ∈ K → g x • f x = 0
[PROOFSTEP]
intros x hx
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
x : X
hx : ¬x ∈ K
⊢ g x • f x = 0
[PROOFSTEP]
simp [image_eq_zero_of_nmem_tsupport hx]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x
⊢ Integrable fun x => g x • f x
[PROOFSTEP]
rw [← this, indicator_smul]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x
⊢ Integrable fun x => g x • Set.indicator K (fun x => f x) x
[PROOFSTEP]
apply Integrable.smul_of_top_right
[GOAL]
case hf
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x
⊢ Integrable fun x => Set.indicator K (fun x => f x) x
[PROOFSTEP]
rw [integrable_indicator_iff hK.measurableSet]
[GOAL]
case hf
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x
⊢ IntegrableOn (fun x => f x) K
[PROOFSTEP]
exact hf.integrableOn_isCompact hK
[GOAL]
case hφ
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
hf : LocallyIntegrable f
g : X → ℝ
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x
⊢ Memℒp (fun x => g x) ⊤
[PROOFSTEP]
exact hg.memℒp_top_of_hasCompactSupport h'g μ
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
⊢ Integrable fun x => f x • g x
[PROOFSTEP]
let K := tsupport g
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
⊢ Integrable fun x => f x • g x
[PROOFSTEP]
have hK : IsCompact K := h'g
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ Integrable fun x => f x • g x
[PROOFSTEP]
have : K.indicator (fun x ↦ f x • g x) = (fun x ↦ f x • g x) :=
  by
  apply indicator_eq_self.2
  apply support_subset_iff'.2
  intros x hx
  simp [image_eq_zero_of_nmem_tsupport hx]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (Set.indicator K fun x => f x • g x) = fun x => f x • g x
[PROOFSTEP]
apply indicator_eq_self.2
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (support fun x => f x • g x) ⊆ K
[PROOFSTEP]
apply support_subset_iff'.2
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ ∀ (x : X), ¬x ∈ K → f x • g x = 0
[PROOFSTEP]
intros x hx
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
x : X
hx : ¬x ∈ K
⊢ f x • g x = 0
[PROOFSTEP]
simp [image_eq_zero_of_nmem_tsupport hx]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => f x • g x) = fun x => f x • g x
⊢ Integrable fun x => f x • g x
[PROOFSTEP]
rw [← this, indicator_smul_left]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => f x • g x) = fun x => f x • g x
⊢ Integrable fun x => Set.indicator K (fun x => f x) x • g x
[PROOFSTEP]
apply Integrable.smul_of_top_left
[GOAL]
case hφ
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => f x • g x) = fun x => f x • g x
⊢ Integrable fun x => Set.indicator K (fun x => f x) x
[PROOFSTEP]
rw [integrable_indicator_iff hK.measurableSet]
[GOAL]
case hφ
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => f x • g x) = fun x => f x • g x
⊢ IntegrableOn (fun x => f x) K
[PROOFSTEP]
exact hf.integrableOn_isCompact hK
[GOAL]
case hf
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (Set.indicator K fun x => f x • g x) = fun x => f x • g x
⊢ Memℒp (fun x => g x) ⊤
[PROOFSTEP]
exact hg.memℒp_top_of_hasCompactSupport h'g μ
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
inst✝¹ : IsLocallyFiniteMeasure μ
K : Set X
a b : X
inst✝ : MetrizableSpace X
hK : IsCompact K
hf : ContinuousOn f K
⊢ IntegrableOn f K
[PROOFSTEP]
letI := metrizableSpaceMetric X
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
inst✝¹ : IsLocallyFiniteMeasure μ
K : Set X
a b : X
inst✝ : MetrizableSpace X
hK : IsCompact K
hf : ContinuousOn f K
this : MetricSpace X := metrizableSpaceMetric X
⊢ IntegrableOn f K
[PROOFSTEP]
refine' LocallyIntegrableOn.integrableOn_isCompact (fun x hx => _) hK
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
inst✝¹ : IsLocallyFiniteMeasure μ
K : Set X
a b : X
inst✝ : MetrizableSpace X
hK : IsCompact K
hf : ContinuousOn f K
this : MetricSpace X := metrizableSpaceMetric X
x : X
hx : x ∈ K
⊢ IntegrableAtFilter f (𝓝[K] x)
[PROOFSTEP]
exact hf.integrableAt_nhdsWithin_of_isSeparable hK.measurableSet hK.isSeparable hx
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
⊢ IntegrableOn f s
[PROOFSTEP]
borelize E
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ IntegrableOn f s
[PROOFSTEP]
obtain rfl | _ := s.eq_empty_or_nonempty
[GOAL]
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
a b : X
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hmono : MonotoneOn f ∅
ha : IsLeast ∅ a
hb : IsGreatest ∅ b
hs : ↑↑μ ∅ ≠ ⊤
h's : MeasurableSet ∅
⊢ IntegrableOn f ∅
[PROOFSTEP]
exact integrableOn_empty
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
⊢ IntegrableOn f s
[PROOFSTEP]
have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
⊢ IntegrableOn f s
[PROOFSTEP]
have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
⊢ IntegrableOn f s
[PROOFSTEP]
have : Metric.Bounded (f '' s) := Metric.bounded_of_bddAbove_of_bddBelow habove hbelow
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Metric.Bounded (f '' s)
⊢ IntegrableOn f s
[PROOFSTEP]
rcases bounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
[GOAL]
case inr.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Metric.Bounded (f '' s)
C : ℝ
hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C
⊢ IntegrableOn f s
[PROOFSTEP]
have A : IntegrableOn (fun _ => C) s μ := by simp only [hs.lt_top, integrableOn_const, or_true_iff]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Metric.Bounded (f '' s)
C : ℝ
hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C
⊢ IntegrableOn (fun x => C) s
[PROOFSTEP]
simp only [hs.lt_top, integrableOn_const, or_true_iff]
[GOAL]
case inr.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Metric.Bounded (f '' s)
C : ℝ
hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C
A : IntegrableOn (fun x => C) s
⊢ IntegrableOn f s
[PROOFSTEP]
refine'
  Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable
    ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
⊢ IntegrableOn f s
[PROOFSTEP]
obtain rfl | h := s.eq_empty_or_nonempty
[GOAL]
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact ∅
hmono : MonotoneOn f ∅
⊢ IntegrableOn f ∅
[PROOFSTEP]
exact integrableOn_empty
[GOAL]
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
h : Set.Nonempty s
⊢ IntegrableOn f s
[PROOFSTEP]
exact
  hmono.integrableOn_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h) hs.measure_lt_top.ne hs.measurableSet
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
⊢ LocallyIntegrable f
[PROOFSTEP]
intro x
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
[GOAL]
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ ∃ a b, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
exact exists_Icc_mem_subset_of_mem_nhds hU
[GOAL]
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
have ab : a ≤ b := xab.1.trans xab.2
[GOAL]
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
ab : a ≤ b
⊢ IntegrableAtFilter f (𝓝 x)
[PROOFSTEP]
refine' ⟨Icc a b, hab, _⟩
[GOAL]
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
ab : a ≤ b
⊢ IntegrableOn f (Icc a b)
[PROOFSTEP]
exact
  (hmono.monotoneOn _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)
    ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
⊢ IntegrableOn (fun x => g x * g' x) A
[PROOFSTEP]
rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A
[PROOFSTEP]
rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1
[PROOFSTEP]
have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ :=
  by
  filter_upwards [ae_restrict_mem hA] with x hx
  refine' (norm_mul_le _ _).trans _
  rw [mul_comm]
  apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖
[PROOFSTEP]
filter_upwards [ae_restrict_mem hA] with x hx
[GOAL]
case h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g x‖
[PROOFSTEP]
refine' (norm_mul_le _ _).trans _
[GOAL]
case h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖
[PROOFSTEP]
rw [mul_comm]
[GOAL]
case h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖
[PROOFSTEP]
apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C
this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖
⊢ Memℒp (fun x => g x * g' x) 1
[PROOFSTEP]
exact Memℒp.of_le_mul hg (hg.aestronglyMeasurable.mul <| (hg'.mono hAK).aestronglyMeasurable hA) this
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : IntegrableOn g' A
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
⊢ IntegrableOn (fun x => g x * g' x) A
[PROOFSTEP]
rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : IntegrableOn g' A
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A
[PROOFSTEP]
rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1
[PROOFSTEP]
have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ :=
  by
  filter_upwards [ae_restrict_mem hA] with x hx
  refine' (norm_mul_le _ _).trans _
  apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖
[PROOFSTEP]
filter_upwards [ae_restrict_mem hA] with x hx
[GOAL]
case h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖
[PROOFSTEP]
refine' (norm_mul_le _ _).trans _
[GOAL]
case h
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖
[PROOFSTEP]
apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
[GOAL]
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C
this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖
⊢ Memℒp (fun x => g x * g' x) 1
[PROOFSTEP]
exact Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ IntegrableOn (fun x => f x • g x) K
[PROOFSTEP]
rw [IntegrableOn, ← integrable_norm_iff]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ Integrable fun a => ‖f a • g a‖
[PROOFSTEP]
simp_rw [norm_smul]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ Integrable fun a => ‖f a‖ * ‖g a‖
[PROOFSTEP]
refine' IntegrableOn.continuousOn_mul _ hg.norm hK
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ ContinuousOn (fun a => ‖f a‖) K
[PROOFSTEP]
exact continuous_norm.comp_continuousOn hf
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)
[PROOFSTEP]
exact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ IntegrableOn (fun x => f x • g x) K
[PROOFSTEP]
rw [IntegrableOn, ← integrable_norm_iff]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ Integrable fun a => ‖f a • g a‖
[PROOFSTEP]
simp_rw [norm_smul]
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ Integrable fun a => ‖f a‖ * ‖g a‖
[PROOFSTEP]
refine' IntegrableOn.mul_continuousOn hf.norm _ hK
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ ContinuousOn (fun a => ‖g a‖) K
[PROOFSTEP]
exact continuous_norm.comp_continuousOn hg
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_5
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)
[PROOFSTEP]
exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : LocallyIntegrableOn f s
hg : ContinuousOn g s
hs : IsOpen s
⊢ LocallyIntegrableOn (fun x => g x * f x) s
[PROOFSTEP]
rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
hg : ContinuousOn g s
hs : IsOpen s
⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => g x * f x) k
[PROOFSTEP]
exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : LocallyIntegrableOn f s
hg : ContinuousOn g s
hs : IsOpen s
⊢ LocallyIntegrableOn (fun x => f x * g x) s
[PROOFSTEP]
rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
hg : ContinuousOn g s
hs : IsOpen s
⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => f x * g x) k
[PROOFSTEP]
exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁵ : OpensMeasurableSpace X
A K : Set X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : T2Space X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : SecondCountableTopologyEither X 𝕜
inst✝ : NormedSpace 𝕜 E
f : X → E
g : X → 𝕜
s : Set X
hs : IsOpen s
hf : LocallyIntegrableOn f s
hg : ContinuousOn g s
⊢ LocallyIntegrableOn (fun x => g x • f x) s
[PROOFSTEP]
rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁵ : OpensMeasurableSpace X
A K : Set X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : T2Space X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : SecondCountableTopologyEither X 𝕜
inst✝ : NormedSpace 𝕜 E
f : X → E
g : X → 𝕜
s : Set X
hs : IsOpen s
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
hg : ContinuousOn g s
⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => g x • f x) k
[PROOFSTEP]
exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁵ : OpensMeasurableSpace X
A K : Set X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : T2Space X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : SecondCountableTopologyEither X E
inst✝ : NormedSpace 𝕜 E
f : X → 𝕜
g : X → E
s : Set X
hs : IsOpen s
hf : LocallyIntegrableOn f s
hg : ContinuousOn g s
⊢ LocallyIntegrableOn (fun x => f x • g x) s
[PROOFSTEP]
rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
[GOAL]
X : Type u_1
Y : Type u_2
E : Type u_3
R : Type u_4
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
f✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁵ : OpensMeasurableSpace X
A K : Set X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : T2Space X
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : SecondCountableTopologyEither X E
inst✝ : NormedSpace 𝕜 E
f : X → 𝕜
g : X → E
s : Set X
hs : IsOpen s
hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k
hg : ContinuousOn g s
⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => f x • g x) k
[PROOFSTEP]
exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c