Theorems · Theorem · probability

ProbabilityTheory.condIndepFun_iff_compProd_map_prod_eq_compProd_prod_map_map

∀ {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω]
  {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'}
  {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'},
  Measurable f →
    Measurable g →
      (ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
        (μ.trim hm').compProd ((ProbabilityTheory.condExpKernel μ m').map fun ω => (f ω, g ω)) =
          (μ.trim hm').compProd
            (((ProbabilityTheory.condExpKernel μ m').map f).prod ((ProbabilityTheory.condExpKernel μ m').map g)))

Two random variables are conditionally independent iff they satisfy the almost sure equality of conditional expectations μ⟦f ⁻¹' s ∩ g ⁻¹' t | m'⟧ =ᵐ[μ] μ⟦f ⁻¹' s | m'⟧ * μ⟦g ⁻¹' t | m'⟧ for all measurable sets s and t (see condIndepFun_iff_condExp_inter_preimage_eq_mul). Here, this is phrased with Markov kernels associated to the conditional expectations, and the almost sure equality is expressed as equality of the composition-product with the measure, which is equivalent to a.e. equality. See condIndepFun_iff_map_prod_eq_prod_map_map for the a.e. equality version with kernels. For a random variable f, (condExpKernel μ m').map f is the law of the conditional expectation of f given m': almost surely, (condExpKernel μ m').map f ω s = μ⟦f ⁻¹' s | m'⟧ ω.

Defined in

Mathlib.Probability.Independence.Conditional

Cited by

2 results in Mathlib

Foundations

Depth 285 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceMeasureTheory.IsFiniteMeasure