(Non-)Distributivity of the Product for $\sigma$-Algebras with Respect to the Intersection

TL;DR

This paper investigates when the distributivity of the product of sigma-algebras over intersection holds, providing counterexamples, conditions for validity, and characterizations involving atoms in specific cases.

Contribution

It offers a counterexample to distributivity, characterizes when it holds for countably generated subspaces, and provides sufficient conditions for distributivity.

Findings

Counterexample for general case

Characterization using atoms in analytic spaces

Sufficient conditions for distributivity

Abstract

We study the validity of the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ where $\mathcal{A}$ is a $\sigma$-algebra on a set $X$, and $\mathcal{F}, \mathcal{G}$ are $\sigma$-algebras on a set $U$. We present a counterexample for the general case and in the case of countably generated subspaces of analytic measurable spaces we give an equivalent condition in terms of the $\sigma$-algebras' atoms. Using this, we give a sufficient condition under which distributivity holds.