Theorem: MeasurableEmbedding.aestronglyMeasurable_map_iff
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g : α → β
⊢ AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ
state_after: α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g : α → β
⊢ AEStronglyMeasurable (g ∘ f) μ → AEStronglyMeasurable g (Measure.map f μ)
tactic: refine ⟨fun H => H.comp_measurable hf.measurable, ?_⟩
state_before: α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g : α → β
⊢ AEStronglyMeasurable (g ∘ f) μ → AEStronglyMeasurable g (Measure.map f μ)
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g : α → β
g₁ : γ → β
hgm₁ : StronglyMeasurable g₁
heq : g ∘ f =ᶠ[ae μ] g₁
⊢ AEStronglyMeasurable g (Measure.map f μ)
tactic: rintro ⟨g₁, hgm₁, heq⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g : α → β
g₁ : γ → β
hgm₁ : StronglyMeasurable g₁
heq : g ∘ f =ᶠ[ae μ] g₁
⊢ AEStronglyMeasurable g (Measure.map f μ)
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g g₂ : α → β
hgm₂ : StronglyMeasurable g₂
hgm₁ : StronglyMeasurable (g₂ ∘ f)
heq : g ∘ f =ᶠ[ae μ] g₂ ∘ f
⊢ AEStronglyMeasurable g (Measure.map f μ)
tactic: rcases hf.exists_stronglyMeasurable_extend hgm₁ fun x => ⟨g x⟩ with ⟨g₂, hgm₂, rfl⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
inst✝² : Countable ι
m : MeasurableSpace α
μ✝ ν : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ✝
f✝ g✝ : α → β
γ : Type u_5
mγ : MeasurableSpace γ
mα : MeasurableSpace α
f : γ → α
μ : Measure γ
hf : MeasurableEmbedding f
g g₂ : α → β
hgm₂ : StronglyMeasurable g₂
hgm₁ : StronglyMeasurable (g₂ ∘ f)
heq : g ∘ f =ᶠ[ae μ] g₂ ∘ f
⊢ AEStronglyMeasurable g (Measure.map f μ)
state_after: no goals
tactic: exact ⟨g₂, hgm₂, hf.ae_map_iff.2 heq⟩