Theorems · Theorem · probability

ProbabilityTheory.iCondIndepFun.condIndepFun_finset

∀ {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {hm' : m' ≤ mΩ}
  {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_6}
  {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (S T : Finset ι),
  Disjoint S T →
    ProbabilityTheory.iCondIndepFun m' hm' f μ →
      (∀ (i : ι), Measurable (f i)) →
        ProbabilityTheory.CondIndepFun m' hm' (fun a i => f (↑i) a) (fun a i => f (↑i) a) μ

If f is a family of mutually conditionally independent random variables (iCondIndepFun m' hm' m f μ) and S, T are two disjoint finite index sets, then the tuple formed by f i for i ∈ S is conditionally independent of the tuple (f i)_i for i ∈ T.

Defined in

Mathlib.Probability.Independence.Conditional

Cited by

0 results in Mathlib

Foundations

Depth 284 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceMeasureTheory.IsFiniteMeasure