-- Mathlib/MeasureTheory/Measure/Prod.lean
theorem skew_product [SFinite μa] [SFinite μc] {f : α → β} (hf : MeasurePreserving f μa μb)
    {g : α → γ → δ} (hgm : Measurable (uncurry g)) (hg : ∀ᵐ a ∂μa, map (g a) μc = μd) :
    MeasurePreserving (fun p : α × γ => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd)

Proof state:
case map_eq.inr.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
⊢ (fun a => μc (Prod.mk a ⁻¹' ((fun p => (f p.1, g p.1 p.2)) ⁻¹' s))) =ᶠ[ae μa] fun a => μd (Prod.mk (f a) ⁻¹' s)

Tactic:
filter_upwards [hg] with a ha

Result:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
a : α
ha : map (g a) μc = μd
⊢ μc (Prod.mk a ⁻¹' ((fun p => (f p.1, g p.1 p.2)) ⁻¹' s)) = μd (Prod.mk (f a) ⁻¹' s)

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
a : α
ha : map (g a) μc = μd
⊢ μc (Prod.mk a ⁻¹' ((fun p => (f p.1, g p.1 p.2)) ⁻¹' s)) = μd (Prod.mk (f a) ⁻¹' s)

Tactic:
rw [← ha, map_apply hgm.of_uncurry_left (measurable_prod_mk_left hs), preimage_preimage,
  preimage_preimage]

Result:
no goals

Proof state:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite (map (g a) μc)

Tactic:
infer_instance

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
⊢ SFinite μd

Tactic:
obtain ⟨a, ha⟩ : ∃ a, map (g a) μc = μd := hg.exists

Result:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite μd

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
⊢ MeasurePreserving (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd)

Tactic:
use this

Result:
case map_eq
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
⊢ map (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) = μb.prod μd

Proof state:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite μd

Tactic:
rw [← ha]

Result:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite (map (g a) μc)

Proof state:
case map_eq
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
⊢ map (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) = μb.prod μd

Tactic:
rcases eq_zero_or_neZero μa with rfl | _

Result:
case map_eq.inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μc
f : α → β
g : α → γ → δ
hgm : Measurable (uncurry g)
this : Measurable fun p => (f p.1, g p.1 p.2)
inst✝ : SFinite 0
hf : MeasurePreserving f 0 μb
hg : ∀ᵐ (a : α) ∂0, map (g a) μc = μd
⊢ map (fun p => (f p.1, g p.1 p.2)) (Measure.prod 0 μc) = μb.prod μd

case map_eq.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p => (f p.1, g p.1 p.2)
h✝ : NeZero μa
⊢ map (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) = μb.prod μd