Theorems · Theorem · probability

ProbabilityTheory.IsCondKernelCDF.setLIntegral

∀ {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} [ProbabilityTheory.IsFiniteKernel κ]
  {f : α × β → StieltjesFunction ℝ},
  ProbabilityTheory.IsCondKernelCDF f κ ν →
    ∀ (a : α) {s : Set β},
      MeasurableSet s → ∀ (x : ℝ), ∫⁻ (b : β) in s, ENNReal.ofReal (↑(f (a, b)) x) ∂ν a = (κ a) (s ×ˢ Set.Iic x)

Defined in

Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Cited by

3 results in Mathlib

Foundations

Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsFiniteKernel