theorem extracted_formal_statement_5.{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' →
    ∀ (ω : Ω),
      AEStronglyMeasurable (fun y => F (f ω, y)) (Measure.map g μ) → StronglyMeasurable fun y => F (f ω, y) := sorry

State before:
case h.h.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'
ω : Ω
⊢ AEStronglyMeasurable (fun y => F (f ω, y)) (Measure.map g μ)

Tactic: apply StronglyMeasurable.aestronglyMeasurable

State after:
[case h.h.h.hfm.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'
 ω : Ω
 ⊢ StronglyMeasurable fun y => F (f ω, y)]

Relevant lemmas:
MeasureTheory.StronglyMeasurable.aestronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β]
  {m m₀ : MeasurableSpace α} {μ : Measure α} {f : α → β}, StronglyMeasurable f → AEStronglyMeasurable f μ