Theorems · Theorem · probability

ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map

∀ {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β}
  {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} [MeasureTheory.IsFiniteMeasure μ],
  AEMeasurable f μ →
    AEMeasurable g μ →
      (ProbabilityTheory.IndepFun f g μ ↔
        MeasureTheory.Measure.map (fun ω => (f ω, g ω)) μ =
          (MeasureTheory.Measure.map f μ).prod (MeasureTheory.Measure.map g μ))

Defined in

Mathlib.Probability.Independence.Basic

Cited by

12 results in Mathlib

Foundations

Depth 223 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure