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

TL;DR

This paper investigates conditions under which the distributivity of the product of sigma algebras over intersection holds or fails, extending previous results and providing new examples and generalizations.

Contribution

It refines conditions for distributivity of sigma algebra products, generalizes prior results, and extends the theory to Blackwell spaces with new counterexamples.

Findings

Distributivity holds under specific refined conditions.

Counterexamples exist for uncountable sets with certain sigma algebra properties.

Extension of results to Blackwell spaces.

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.