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This question already has an answer here:

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?

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

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    $\begingroup$ What did you try? Which partial results can you prove? $\endgroup$ – Did Apr 15 '13 at 20:11
  • $\begingroup$ $\supset$ is the easy one $\endgroup$ – user72739 Apr 15 '13 at 20:22
  • $\begingroup$ You could include it in the OP. Also, I think you can remove the tag (stochastic-calculus). $\endgroup$ – Davide Giraudo Apr 15 '13 at 20:56
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    $\begingroup$ sorry, I just found this math.stackexchange.com/questions/92546/… anyway, thx $\endgroup$ – user72739 Apr 15 '13 at 20:59

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