Theorems · Theorem · probability

ProbabilityTheory.Kernel.setLIntegral_compProd_univ_left

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ]
  (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ENNReal},
  Measurable f →
    ∀ {t : Set γ},
      MeasurableSet t →
        ∫⁻ (z : β × γ) in Set.univ ×ˢ t, f z ∂(κ.compProd η) a = ∫⁻ (x : β), ∫⁻ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a

Defined in

Mathlib.Probability.Kernel.Composition.CompProd

Cited by

0 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernel