Theorems · Theorem · probability

ProbabilityTheory.Kernel.indepFun_iff_indepSet_preimage

∀ {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω}
  {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β}
  {mβ' : MeasurableSpace β'} [ProbabilityTheory.IsZeroOrMarkovKernel κ],
  Measurable f →
    Measurable g →
      (ProbabilityTheory.Kernel.IndepFun f g κ μ ↔
        ∀ (s : Set β) (t : Set β'),
          MeasurableSet s → MeasurableSet t → ProbabilityTheory.Kernel.IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ)

Defined in

Mathlib.Probability.Independence.Kernel.IndepFun

Cited by

2 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsZeroOrMarkovKernel