# Interchange product $\sigma$-algebra and countable intersection [duplicate]

If $(\mathcal F_n)_{n\in\mathbb N}$ is a decreasing sequence of $\sigma$-algebras and $\mathcal G$ another $\sigma$-algebra, is it possible to interchange the intersection with the product $\sigma$-algebra?

$\bigcap\limits_{n\in \mathbb N}(\mathcal F_n\otimes \mathcal G)=(\bigcap\limits_{n\in \mathbb N}\mathcal F_n\otimes \mathcal G)$

If not, is there a condition for this equality?

## marked as duplicate by Davide Giraudo, Amzoti, Stefan Hansen, Micah, rschwiebApr 16 '13 at 17:03

• $\supset$ is the easy one – user72739 Apr 15 '13 at 20:22