Theorems · Theorem · probability

ProbabilityTheory.condIndepFun_iff_condIndepSet_preimage

∀ {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω]
  {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'}
  {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'},
  Measurable f →
    Measurable g →
      (ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
        ∀ (s : Set β) (t : Set β'),
          MeasurableSet s → MeasurableSet t → ProbabilityTheory.CondIndepSet m' hm' (f ⁻¹' s) (g ⁻¹' t) μ)

Defined in

Mathlib.Probability.Independence.Conditional

Cited by

0 results in Mathlib

Foundations

Depth 285 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceMeasureTheory.IsFiniteMeasure