Theorems · Theorem · probability

ProbabilityTheory.indepFun_prod

∀ {Ω : Type u_10} {Ω' : Type u_11} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν]
  {𝓧 : Type u_12} {𝓨 : Type u_13} [inst : MeasurableSpace 𝓧] [inst_1 : MeasurableSpace 𝓨] {X : Ω → 𝓧} {Y : Ω' → 𝓨},
  Measurable X → Measurable Y → ProbabilityTheory.IndepFun (fun ω => X ω.1) (fun ω => Y ω.2) (μ.prod ν)

Given random variables X : Ω → 𝓧 and Y : Ω' → 𝓨, they are independent when viewed as random variables defined on the product space Ω × Ω'.

Defined in

Mathlib.Probability.Independence.Basic

Cited by

3 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasureMeasureTheory.IsProbabilityMeasureMeasurableSpaceMeasurableSpace