Theorems · Theorem · probability

ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd_of_measurable

∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω]
  [inst_2 : Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst_3 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β},
  Measurable X →
    Measurable Y →
      ∀ {κ : ProbabilityTheory.Kernel β Ω} [ProbabilityTheory.IsFiniteKernel κ],
        MeasureTheory.Measure.map (fun x => (X x, Y x)) μ = (MeasureTheory.Measure.map X μ).compProd κ →
          ⇑(ProbabilityTheory.condDistrib Y X μ) =ᵐ[MeasureTheory.Measure.map X μ] ⇑κ

condDistrib is a.e. uniquely defined as the kernel satisfying the defining property of condKernel.

Defined in

Mathlib.Probability.Kernel.CondDistrib

Cited by

2 results in Mathlib

Foundations

Depth 282 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceStandardBorelSpaceNonemptyMeasureTheory.IsFiniteMeasureProbabilityTheory.IsFiniteKernel