I am talking abount the product sigma algebra of two algebras. I think we had the statement in the lecture so it should be true. I can not remember the proof completely though and need some help to find it.
Let $\mathcal A$ be the set of all countable unions of disjoint rectangles with measurable sides. It suffices to show that $\mathcal A$ is a $\sigma$-algebra. The step that I can not get right though is showing that $\mathcal A$ is closed under complementation. As far as I remember the proof went like this: Let $A=\bigcup_iR_i$ be a countable union of disjoint rectangles. Then $A^\complement=\bigcap_iR_i^\complement$. I can show that $R_i^\complement\in\mathcal A$ and that finite intersections of them are in $\mathcal A$ as well. But I don't know why countable intersections of the $R_i^\complement$s are supposed to be in $\mathcal A$.
I am not 100% sure if the proof really went like this though.