Theorems · Theorem · probability

ProbabilityTheory.Kernel.compProd_apply_prod

∀ {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} {η : ProbabilityTheory.Kernel (α × β) γ}
  [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] {a : α} {s : Set β} {t : Set γ},
  MeasurableSet s → MeasurableSet t → ((κ.compProd η) a) (s ×ˢ t) = ∫⁻ (b : β) in s, (η (a, b)) t ∂κ a

Defined in

Mathlib.Probability.Kernel.Composition.CompProd

Cited by

3 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernel