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 $σ$-algebra on a set $X$, and $\mathcal{F}, \mathcal{G}$ are $σ$-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 $σ$-algebras' atoms. Using this, we give a sufficient condition under which distributivity holds.