Theorems · Theorem · probability

ProbabilityTheory.Kernel.iIndepFun.indepFun_finset

∀ {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω}
  {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8}
  {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (S T : Finset ι),
  Disjoint S T →
    ProbabilityTheory.Kernel.iIndepFun f κ μ →
      (∀ (i : ι), Measurable (f i)) → ProbabilityTheory.Kernel.IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ μ

If f is a family of mutually independent random variables (iIndepFun m f μ) and S, T are two disjoint finite index sets, then the tuple formed by f i for i ∈ S is independent of the tuple (f i)_i for i ∈ T.

Defined in

Mathlib.Probability.Independence.Kernel.IndepFun

Cited by

8 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound