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My question is relatively simple: what is a product $\sigma$-algebra? And why they are important?

Can anyone suggest any links of intuitive (possibly with simple figures) explanations? Or, maybe someone can explain it. Thanks.

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up vote 3 down vote accepted

They are important because they are necessary for the most correct statements of the Tonelli and Fubini theorems.

I'm not sure there is a natural way of thinking about it. Do you have an intuition for how open sets in $X \times Y$ look when you know what basic open sets in $X$ and $Y$ look like? Even though basic open sets in $X \times Y$ are "rectangles", there can still be "open disks" in $X \times Y$.

The situation for $\sigma$-algebras is similar, but even uglier. That is the product $\sigma$-algebra $\mathbb E \otimes \mathbb F$ on $X \times Y$ when $(X,\mathbb E)$ and $(Y,\mathbb F)$ are measurable spaces is the smallest $\sigma$-algebra containing $\{A \times B \subseteq X\times Y \mathrel| A \in\mathbb E,\thinspace B \in \mathbb F\}$.

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