∀ {ι : Type u_1} {α : ι → Type u_2} {mα : (i : ι) → MeasurableSpace (α i)}
  {P : (J : Finset ι) → MeasureTheory.Measure ((j : ↥J) → α ↑j)} {s : Set ((i : ι) → α i)} {I : Finset ι}
  {S : Set ((i : ↥I) → α ↑i)},
  MeasureTheory.IsProjectiveMeasureFamily P →
    s ∈ MeasureTheory.measurableCylinders α →
      s = MeasureTheory.cylinder I S → MeasurableSet S → MeasureTheory.projectiveFamilyFun P s = (P I) S