  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.
