Complements and Improvements Regarding Distributivity of the Product for $\sigma$-Algebras with Respect to the Intersection

Abstract

We present a variety of refined conditions for $\sigma$ algebras $\mathcal{A}$ (on a set $X$), $\mathcal{F}, \mathcal{G}$ (on a set $U$) such that the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ holds -- or is violated. \\ The article generalizes the results in arXiv:2007.06095 and includes a positive result for $\sigma$ algebras generated by at most countable partitions, was not covered before. We also present a proof that counterexamples may be constructed whenever $X$ is uncountable and there exist two $\sigma$-algebras on $X$ which are both countably separated, but their intersection is not. We present examples of such structures. In the last section, we extend Theorem 3.3 of arXiv:2007.06095 from analytic to the setting of Blackwell spaces.