∀ {Ω : Type u_1} {S : Type u_2} {m mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} {E : S → Type u_4}
  [inst : (s : S) → TopologicalSpace (E s)] [inst_1 : (s : S) → MeasurableSpace (E s)] [∀ (s : S), BorelSpace (E s)]
  [∀ (s : S), HasOuterApproxClosed (E s)] {X : (s : S) → Ω → E s} [MeasureTheory.IsProbabilityMeasure P],
  m ≤ mΩ →
    (∀ (s : S), Measurable (X s)) →
      (∀ (A : Set Ω),
          MeasurableSet A →
            ∀ (I : Finset S) (f : (s : ↥I) → BoundedContinuousFunction (E ↑s) ℝ),
              ∫ (ω : Ω) in A, ∏ s, (f s) (X (↑s) ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X (↑s) ω) ∂P) →
        ProbabilityTheory.Indep m (MeasurableSpace.comap (fun ω s => X s ω) MeasurableSpace.pi) P