Theorem: MeasureTheory.Measure.map_congr
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: by_cases hf : AEMeasurable f μ
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
⊢ map f μ = map g μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
⊢ map f μ = map g μ

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : ¬AEMeasurable f μ
⊢ map f μ = map g μ

tactic: have hg : AEMeasurable g μ := hf.congr h
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
⊢ map f μ = map g μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ map f μ = map g μ

tactic: simp only [← mapₗ_mk_apply_of_aemeasurable hf, ← mapₗ_mk_apply_of_aemeasurable hg]
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ map f μ = map g μ
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ (mapₗ (AEMeasurable.mk f hf)) μ = (mapₗ (AEMeasurable.mk g hg)) μ

tactic: exact
  mapₗ_congr hf.measurable_mk hg.measurable_mk (hf.ae_eq_mk.symm.trans (h.trans hg.ae_eq_mk))
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ (mapₗ (AEMeasurable.mk f hf)) μ = (mapₗ (AEMeasurable.mk g hg)) μ
state_after: no goals

tactic: have hg : ¬AEMeasurable g μ := by simpa [← aemeasurable_congr h] using hf
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : ¬AEMeasurable f μ
⊢ map f μ = map g μ
state_after: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : ¬AEMeasurable f μ
hg : ¬AEMeasurable g μ
⊢ map f μ = map g μ

tactic: simp [map_of_not_aemeasurable, hf, hg]
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : ¬AEMeasurable f μ
hg : ¬AEMeasurable g μ
⊢ map f μ = map g μ
state_after: no goals

tactic: simpa [← aemeasurable_congr h] using hf
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f g : α → β
h : f =ᶠ[ae μ] g
hf : ¬AEMeasurable f μ
⊢ ¬AEMeasurable g μ
state_after: no goals