theorem extracted_formal_statement_1.{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' →
    ∫ (ω : Ω), ∫ (ω' : Ω), F (f ω, g ω') ∂μ ∂μ = ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ →
      (fun a => ∫ (ω' : Ω), F (f a, g ω') ∂μ) =ᶠ[ae μ] fun a => ∫ (x' : β'), F (f a, x') ∂ν' := sorry

State before:
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'
⊢ ∫ (ω : Ω), ∫ (ω' : Ω), F (f ω, g ω') ∂μ ∂μ = ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ

Tactic: apply integral_congr_ae

State after:
[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'
 ⊢ (fun a => ∫ (ω' : Ω), F (f a, g ω') ∂μ) =ᶠ[ae μ] fun a => ∫ (x' : β'), F (f a, x') ∂ν']

Relevant lemmas:
MeasureTheory.integral_congr_ae : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G]
  [inst_3 : NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} {f g : α → G},
  f =ᶠ[ae μ] g → ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ