Theorems · Theorem · probability

ProbabilityTheory.Kernel.fst_compProd_apply

∀ {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] (x : α) {s : Set β},
  MeasurableSet s → ((κ.compProd η).fst x) s = ∫⁻ (b : β), s.indicator (fun b => (η (x, b)) Set.univ) b ∂κ x

If η is a Markov kernel, use instead fst_compProd to get (κ ⊗ₖ η).fst = κ.

Defined in

Mathlib.Probability.Kernel.Composition.CompProd

Cited by

2 results in Mathlib

Foundations

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

Assumes

ProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernel