(Non-)Distributivity of the product for $$\sigma $$-algebras with respect to the intersection

We study the validity of the distributivity equation $$\begin{aligned} ({\mathcal {A}}\otimes {\mathcal {F}})\cap ({\mathcal {A}}\otimes {\mathcal {G}}) ={\mathcal {A}}\otimes \left( {\mathcal {F}}\cap {\mathcal {G}}\right) , \end{aligned}$$ ( A ⊗ F ) ∩ ( A ⊗ G ) = A ⊗ F ∩ G , where $${\mathcal {A}}$$ A is a $$\sigma $$ σ -algebra on a set X, and $${\mathcal {F}}, {\mathcal {G}}$$ F , G are $$\sigma $$ 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 $$\sigma $$ σ -algebras’ atoms. Using this, we give a sufficient condition under which distributivity holds.