Theorems · Theorem · probability

ProbabilityTheory.Kernel.compProd_assoc

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} {δ : Type u_4} {mδ : MeasurableSpace δ} {κ : ProbabilityTheory.Kernel α β}
  {η : ProbabilityTheory.Kernel (α × β) γ} {ξ : ProbabilityTheory.Kernel (α × β × γ) δ},
  (κ.compProd (η.compProd (ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯))).map ⇑MeasurableEquiv.prodAssoc.symm =
    (κ.compProd η).compProd ξ

Kernel.compProd is associative. We have to insert MeasurableEquiv.prodAssoc in two places because the products of types α × β × γ and (α × β) × γ are different.

Defined in

Mathlib.Probability.Kernel.Composition.CompProd

Cited by

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Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound