theorem extracted_formal_statement_16.{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' = ∫ (z : β × β'), F z ∂Measure.map (fun ω => (f ω, g ω)) μ := 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'
⊢ ∫ (z : β × β'), F z ∂ν.prod ν' = ∫ (z : β × β'), F z ∂Measure.map (fun ω => (f ω, g ω)) μ

Tactic: rw [← hfg']

State after:

Relevant lemmas: