-- Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s)

Proof state:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ ∀ (i : ℕ), AEStronglyMeasurable f (μ.restrict (u i ∩ s))

Tactic:
exact fun i : ℕ => (hu i).aestronglyMeasurable

Result:
no goals

Proof state:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
rw [this, aestronglyMeasurable_iUnion_iff]

Result:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ ∀ (i : ℕ), AEStronglyMeasurable f (μ.restrict (u i ∩ s))

Proof state:
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ s = (⋃ i, u i) ∩ s

Tactic:
exact (inter_eq_right.mpr su).symm

Result:
no goals

Proof state:
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ s = ⋃ n, u n ∩ s

Tactic:
rw [← iUnion_inter]

Result:
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ s = (⋃ i, u i) ∩ s

Proof state:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right.mpr su).symm

Result:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ AEStronglyMeasurable f (μ.restrict s)

Proof state:
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
rcases hf.exists_nat_integrableOn with ⟨u, -, su, hu⟩

Result:
case intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ AEStronglyMeasurable f (μ.restrict s)