Theorems · Theorem · probability

indepSets_comap_process_of_prod_bcf

∀ {Ω : Type u_1} {S : Type u_2} {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]
  {𝒜 : Set (Set Ω)},
  (∀ A ∈ 𝒜, MeasureTheory.NullMeasurableSet A P) →
    (∀ (s : S), AEMeasurable (X s) P) →
      (∀ A ∈ 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.IndepSets 𝒜 {A | MeasurableSet A} P

Defined in

Mathlib.Probability.Independence.BoundedContinuousFunction

Cited by

1 results in Mathlib

Foundations

Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMeasurableSpaceBorelSpaceHasOuterApproxClosedMeasureTheory.IsProbabilityMeasure