theorem extracted_formal_statement_14.{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' →
    ∫ (z : β × β'), F z ∂ν.prod ν' = ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν → Integrable F (ν.prod ν') := 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'
⊢ ∫ (z : β × β'), F z ∂ν.prod ν' = ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν

Tactic: apply MeasureTheory.integral_prod

State after:
[case h.hf
 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'
 ⊢ Integrable F (ν.prod ν')]

Relevant lemmas:
MeasureTheory.integral_prod : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α]
  [inst_3 : MeasurableSpace β] {μ : Measure α} {ν : Measure β} [inst_4 : NormedAddCommGroup E] [inst_5 : SFinite ν]
  [inst_6 : NormedSpace ℝ E] [inst_7 : SFinite μ] (f : α × β → E),
  Integrable f (μ.prod ν) → ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ