Theorems · Theorem · probability

ProbabilityTheory.indepFun_iff_indepSet_preimage

∀ {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.IsZeroOrProbabilityMeasure μ],
  Measurable f →
    Measurable g →
      (ProbabilityTheory.IndepFun f g μ ↔
        ∀ (s : Set β) (t : Set β'),
          MeasurableSet s → MeasurableSet t → ProbabilityTheory.IndepSet (f ⁻¹' s) (g ⁻¹' t) μ)

Defined in

Mathlib.Probability.Independence.Basic

Cited by

0 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsZeroOrProbabilityMeasure