Decreasing sequence of product $\sigma$-algebras My question is as follows. 

If $(X,\mathcal A)$ and $(Y,\mathcal B)$ are two measure spaces and $\mathcal A_n\subseteq \mathcal A$ is a decreasing sequence of $\sigma$-algebras with $\mathcal A_n\downarrow \mathcal A_\infty$, then we have 
  $$\mathcal A_n \times \mathcal B\downarrow \mathcal A_\infty \times \mathcal B$$

One direction is easy, but I am stuck at the other direction. Thank you for helping me with this.
I only did one direction as follows: take $A\times B\in \mathcal A_\infty \times \mathcal B$, we have that $A\times B\in \mathcal A_n\times \mathcal B$ for all $n$. So, $A\times B\in \bigcap_n(\mathcal A_n \times \mathcal B)$. The latter is a $\sigma$-algebra, so it contains $\mathcal A_\infty \times \mathcal B$, which is generated by the class of sets of the form $A\times B$. This implies $\mathcal A_\infty \times \mathcal B \subseteq \bigcap_n(\mathcal A_n \times \mathcal B)$.
We need to prove that $\bigcap_n(\mathcal A_n \times \mathcal B)\subseteq \mathcal A_\infty \times \mathcal B$.
Update: I should add another condition, that both $(X,\mathcal A,\mu)$ and $(Y,\mathcal B,\nu)$ are probability spaces, and all the $\sigma-$ algebras are considered modulo the null sets. But I still want to know what happens if we don't have such a condition. Are there any counterexamples?

If $(X,{\mathcal A},P)$ and $(Y,{\mathcal B}, Q)$ are probability spaces, then
$\cap_n({\mathcal A}_n\times{\mathcal B})$ is contained in the $P\times Q$-completion of ${\mathcal A}_\infty\times {\mathcal B}$.  One way to prove this is by the martingale convergence theorem. (Exercise 6 on page 71 of Parry's book refers to the product of probability spaces, not just measurable spaces. Parry notes on page 8 of his text that inclusion between sigma-algebras is to be understood modulo null sets.)
