Theorem: MeasureTheory.pdf.quasiMeasurePreserving_hasPDF
File: Mathlib/Probability/Density.lean

tactic: wlog hmX : Measurable X
state_before: Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
⊢ HasPDF (g ∘ X) ℙ ν
state_after: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
⊢ HasPDF (g ∘ X) ℙ ν

Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ HasPDF (g ∘ X) ℙ ν

tactic: rw [hasPDF_iff, ← map_map hg.measurable hmX]
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ HasPDF (g ∘ X) ℙ ν
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ AEMeasurable (g ∘ X) ℙ ∧ (map g (map X ℙ)).HaveLebesgueDecomposition ν ∧ map g (map X ℙ) ≪ ν

tactic: refine ⟨(hg.measurable.comp hmX).aemeasurable, hmap, ?_⟩
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ AEMeasurable (g ∘ X) ℙ ∧ (map g (map X ℙ)).HaveLebesgueDecomposition ν ∧ map g (map X ℙ) ≪ ν
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ map g (map X ℙ) ≪ ν

tactic: rw [map_eq_withDensity_pdf X ℙ μ]
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ map g (map X ℙ) ≪ ν
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ map g (μ.withDensity (pdf X ℙ μ)) ≪ ν

tactic: refine AbsolutelyContinuous.mk fun s hsm hs => ?_
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
⊢ map g (μ.withDensity (pdf X ℙ μ)) ≪ ν
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
⊢ (map g (μ.withDensity (pdf X ℙ μ))) s = 0

tactic: rw [map_apply hg.measurable hsm, withDensity_apply _ (hg.measurable hsm)]
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
⊢ (map g (μ.withDensity (pdf X ℙ μ))) s = 0
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0

tactic: have := hg.absolutelyContinuous hs
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
this : (map g μ) s = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0

tactic: rw [map_apply hg.measurable hsm] at this
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
this : (map g μ) s = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0
state_after: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
this : μ (g ⁻¹' s) = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0

tactic: exact setLIntegral_measure_zero _ _ this
state_before: Ω✝ : Type u_1
E✝ : Type u_2
inst✝⁴ : MeasurableSpace E✝
m✝ : MeasurableSpace Ω✝
ℙ✝ : Measure Ω✝
μ✝ : Measure E✝
F✝ : Type u_3
inst✝³ : MeasurableSpace F✝
ν✝ : Measure F✝
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
hmX : Measurable X
s : Set F
hsm : MeasurableSet s
hs : ν s = 0
this : μ (g ⁻¹' s) = 0
⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ μ a ∂μ = 0
state_after: no goals

tactic: have hae : g ∘ X =ᵐ[ℙ] g ∘ hX.mk := hX.ae_eq_mk.mono fun x h ↦ by dsimp; rw [h]
state_before: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
⊢ HasPDF (g ∘ X) ℙ ν
state_after: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
⊢ HasPDF (g ∘ X) ℙ ν

tactic: have hXmk : HasPDF hX.mk ℙ μ := HasPDF.congr hX.ae_eq_mk
state_before: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
⊢ HasPDF (g ∘ X) ℙ ν
state_after: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
hXmk : HasPDF (AEMeasurable.mk X hX) ℙ μ
⊢ HasPDF (g ∘ X) ℙ ν

tactic: apply (HasPDF.congr' hae).mpr
state_before: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
hXmk : HasPDF (AEMeasurable.mk X hX) ℙ μ
⊢ HasPDF (g ∘ X) ℙ ν
state_after: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
hXmk : HasPDF (AEMeasurable.mk X hX) ℙ μ
⊢ HasPDF (g ∘ AEMeasurable.mk X hX) ℙ ν

tactic: exact this hX.measurable_mk.aemeasurable hg (map_congr hX.ae_eq_mk ▸ hmap) hX.measurable_mk
state_before: case inr
Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
hae : g ∘ X =ᶠ[ae ℙ] g ∘ AEMeasurable.mk X hX
hXmk : HasPDF (AEMeasurable.mk X hX) ℙ μ
⊢ HasPDF (g ∘ AEMeasurable.mk X hX) ℙ ν
state_after: no goals

tactic: dsimp
state_before: Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
x : Ω
h : X x = AEMeasurable.mk X hX x
⊢ (g ∘ X) x = (g ∘ AEMeasurable.mk X hX) x
state_after: Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
x : Ω
h : X x = AEMeasurable.mk X hX x
⊢ g (X x) = g (AEMeasurable.mk X hX x)

tactic: rw [h]
state_before: Ω : Type u_1
E : Type u_2
inst✝² : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
F : Type u_3
inst✝¹ : MeasurableSpace F
ν : Measure F
X : Ω → E
inst✝ : HasPDF X ℙ μ
hX : AEMeasurable X ℙ
g : E → F
hg : QuasiMeasurePreserving g μ ν
hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν
this :
  ∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
    {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : Measure F} {X : Ω → E} [inst_2 : HasPDF X ℙ μ],
    AEMeasurable X ℙ →
      ∀ {g : E → F},
        QuasiMeasurePreserving g μ ν → (map g (map X ℙ)).HaveLebesgueDecomposition ν → Measurable X → HasPDF (g ∘ X) ℙ ν
hmX : ¬Measurable X
x : Ω
h : X x = AEMeasurable.mk X hX x
⊢ g (X x) = g (AEMeasurable.mk X hX x)
state_after: no goals