Theorems · Theorem · probability

ProbabilityTheory.setLIntegral_toKernel_Iic

∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction ℝ}
  [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsCondKernelCDF f κ ν) (a : α) (x : ℝ) {s : Set β},
  MeasurableSet s →
    ∫⁻ (b : β) in s, ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) (Set.Iic x) ∂ν a =
      (κ a) (s ×ˢ Set.Iic x)

Defined in

Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Cited by

2 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsFiniteKernel