Theorem: MeasureTheory.QuasiMeasurePreserving.prod_of_right
File: Mathlib/MeasureTheory/Constructions/Prod/Basic.lean

Proof steps:
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
⊢ QuasiMeasurePreserving f (μ.prod ν) τ
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
⊢ map f (μ.prod ν) ≪ τ
tactic: refine ⟨hf, ?_⟩
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
⊢ map f (μ.prod ν) ≪ τ
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ (map f (μ.prod ν)) s = 0
tactic: refine AbsolutelyContinuous.mk fun s hs h2s => ?_
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ (map f (μ.prod ν)) s = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' (f ⁻¹' s)) ∂μ = 0
tactic: rw [map_apply hf hs, prod_apply (hf hs)]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' (f ⁻¹' s)) ∂μ = 0
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ = 0
tactic: simp_rw [preimage_preimage]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ✝ : Measure γ
inst✝¹ : NormedAddCommGroup E
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ = 0
state_after: no goals
tactic: rw [lintegral_congr_ae (h2f.mono fun x hx => hx.preimage_null h2s), lintegral_zero]