Theorems · Theorem · probability

ProbabilityTheory.condIndepFun_iff_map_prod_eq_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'} [MeasurableSpace.CountableOrCountablyGenerated Ω (β × β')],
  Measurable f →
    Measurable g →
      (ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
        ⇑((ProbabilityTheory.condExpKernel μ m').map fun ω => (f ω, g ω)) =ᵐ[μ.trim hm']
          ⇑(((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. 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

0 results in Mathlib

Foundations

Depth 338 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceMeasureTheory.IsFiniteMeasureMeasurableSpace.CountableOrCountablyGenerated