theorem extracted_formal_statement_10.{u_4, u_3, u_2, u_1} {E : Type u_1} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] {Ω : Type u_2} {β : Type u_3} {β' : Type u_4} {_mΩ : MeasurableSpace Ω} {μ : Measure Ω}
  {f : Ω → β} {g : Ω → β'} {F : β × β' → E} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
  [inst_2 : IsFiniteMeasure μ] {hF' : StronglyMeasurable F} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
  (hfg : IndepFun f g μ) (hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))) :
  let ν := Measure.map f μ;
  let ν' := Measure.map g μ;
  Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν' →
    ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν = ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ →
      AEStronglyMeasurable (fun y => ∫ (x' : β'), F (y, x') ∂ν') (Measure.map f μ) := sorry

State before:
case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
Ω : Type u_2
β : Type u_3
β' : Type u_4
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → β
g : Ω → β'
F : β × β' → E
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsFiniteMeasure μ
hF' : StronglyMeasurable F
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfg : IndepFun f g μ
hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))
ν : Measure β := Measure.map f μ
ν' : Measure β' := Measure.map g μ
hfg' : Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν'
⊢ ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν = ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ

Tactic: apply MeasureTheory.integral_map

State after:
[case h.hφ
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : NormedSpace ℝ E
 Ω : Type u_2
 β : Type u_3
 β' : Type u_4
 _mΩ : MeasurableSpace Ω
 μ : Measure Ω
 f : Ω → β
 g : Ω → β'
 F : β × β' → E
 mβ : MeasurableSpace β
 mβ' : MeasurableSpace β'
 inst✝ : IsFiniteMeasure μ
 hF' : StronglyMeasurable F
 hf : AEMeasurable f μ
 hg : AEMeasurable g μ
 hfg : IndepFun f g μ
 hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))
 ν : Measure β := Measure.map f μ
 ν' : Measure β' := Measure.map g μ
 hfg' : Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν'
 ⊢ AEMeasurable f μ,
 case h.hfm
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : NormedSpace ℝ E
 Ω : Type u_2
 β : Type u_3
 β' : Type u_4
 _mΩ : MeasurableSpace Ω
 μ : Measure Ω
 f : Ω → β
 g : Ω → β'
 F : β × β' → E
 mβ : MeasurableSpace β
 mβ' : MeasurableSpace β'
 inst✝ : IsFiniteMeasure μ
 hF' : StronglyMeasurable F
 hf : AEMeasurable f μ
 hg : AEMeasurable g μ
 hfg : IndepFun f g μ
 hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))
 ν : Measure β := Measure.map f μ
 ν' : Measure β' := Measure.map g μ
 hfg' : Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν'
 ⊢ AEStronglyMeasurable (fun y => ∫ (x' : β'), F (y, x') ∂ν') (Measure.map f μ)]

Relevant lemmas:
MeasureTheory.integral_map : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_3 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [inst_4 : MeasurableSpace β] {φ : α → β},
  AEMeasurable φ μ →
    ∀ {f : β → G}, AEStronglyMeasurable f (Measure.map φ μ) → ∫ (y : β), f y ∂Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ