Product sigma-algebra

Remarks

For two factors, $\Sigma_1 \otimes \Sigma_2$ is generated by the rectangles $A_1 \times A_2$ with $A_1 \in \Sigma_1$ and $A_2 \in \Sigma_2$, since $A_1 \times A_2 = \pi_1^{-1}(A_1) \cap \pi_2^{-1}(A_2)$. For general $I$, it is generated by the cylinder sets $\prod_{i \in J} A_i \times \prod_{i \notin J} X_i$ with $J \subseteq I$ finite.

Example: $\mathcal{B}(\R) \otimes \mathcal{B}(\R) = \mathcal{B}(\R^2)$, where $\mathcal{B}$ denotes the Borel σ-algebra. Each rectangle of Borel sets is Borel in $\R^2$, which gives one inclusion. Each open set of $\R^2$ is a countable union of open rectangles with rational corners, which gives the other.

Non-example: let $\mathcal{L}$ be the σ-algebra of Lebesgue measurable subsets of $\R$. Then $\mathcal{L} \otimes \mathcal{L}$ is strictly smaller than the σ-algebra of Lebesgue measurable subsets of $\R^2$. For a non-measurable $N \subseteq \R$, the set $\set{0} \times N$ is a Lebesgue null set in $\R^2$, but its section at $0$ is $N \notin \mathcal{L}$, and every set in $\mathcal{L} \otimes \mathcal{L}$ has measurable sections.

A function $f \colon Z \to X$ is measurable for $\bigotimes_i \Sigma_i$ if and only if every coordinate $\pi_i \circ f$ is measurable.