theorem extracted_formal_statement_0.{u_4, u_3, u_2, u_1} {E : Type u_1} [inst : NormedAddCommGroup 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_1 : 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' := 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
⊢ Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν'

Tactic: exact (ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map hf hg).mp hfg

State after:

Relevant lemmas:
ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map : ∀ {Ω : Type u_1} {β : Type u_6} {β' : Type u_7}
  {_mΩ : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
  [inst : IsFiniteMeasure μ],
  AEMeasurable f μ →
    AEMeasurable g μ → (IndepFun f g μ ↔ Measure.map (fun ω => (f ω, g ω)) μ = (Measure.map f μ).prod (Measure.map g μ))