Theorems · Theorem · probability

ProbabilityTheory.Kernel.iIndepFun_iff_measure_inter_preimage_eq_mul

∀ {α : Type u_1} {Ω : Type u_2} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω}
  {μ : MeasureTheory.Measure α} {ι : Type u_8} {β : ι → Type u_9} (m : (x : ι) → MeasurableSpace (β x))
  (f : (i : ι) → Ω → β i),
  ProbabilityTheory.Kernel.iIndepFun f κ μ ↔
    ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) →
        ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)

Defined in

Mathlib.Probability.Independence.Kernel.IndepFun

Cited by

5 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound