Theorem: MeasureTheory.Measure.restrict_map_of_aemeasurable
File: Mathlib/MeasureTheory/Measure/AEMeasurable.lean

tactic: congr 1
state_before: ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
⊢ restrict (map f μ) s = restrict (map (AEMeasurable.mk f hf) μ) s

tactic: apply Measure.map_congr hf.ae_eq_mk
state_before: case e_μ
ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
⊢ map f μ = map (AEMeasurable.mk f hf) μ

tactic: apply congr_arg
state_before: ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
⊢ map f (restrict μ (AEMeasurable.mk f hf ⁻¹' s)) = map f (restrict μ (f ⁻¹' s))

tactic: ext1 t ht
state_before: case h
ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
⊢ restrict μ (AEMeasurable.mk f hf ⁻¹' s) = restrict μ (f ⁻¹' s)

tactic: simp only [ht, Measure.restrict_apply]
state_before: case h.h
ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
⊢ ↑↑(restrict μ (AEMeasurable.mk f hf ⁻¹' s)) t = ↑↑(restrict μ (f ⁻¹' s)) t

tactic: apply measure_congr
state_before: case h.h
ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
⊢ ↑↑μ (t ∩ AEMeasurable.mk f hf ⁻¹' s) = ↑↑μ (t ∩ f ⁻¹' s)

tactic: apply (EventuallyEq.refl _ _).inter (hf.ae_eq_mk.symm.preimage s)
state_before: case h.h.H
ι : Type ?u.4969569
α : Type u_1
β : Type ?u.4969575
γ : Type ?u.4969578
δ : Type u_2
R : Type ?u.4969584
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
inst✝¹ : MeasurableSpace δ
f✝ g : α → β
μ ν : Measure α
inst✝ : Zero β
f : α → δ
hf : AEMeasurable f
s : Set δ
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
⊢ t ∩ AEMeasurable.mk f hf ⁻¹' s =ᶠ[ae μ] t ∩ f ⁻¹' s