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

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.