Definition:Product Sigma-Algebra/Binary Case

Definition
Let $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ be measurable spaces.

The product $\sigma$-algebra of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as:


 * $\Sigma_1 \otimes \Sigma_2 := \sigma \left({\Sigma_1 \times \Sigma_2}\right)$

where $\sigma$ denotes generated $\sigma$-algebra, and $\times$ denotes Cartesian product.

This product $\sigma$-algebra $\Sigma_1 \otimes \Sigma_2$ is a $\sigma$-algebra on $X \times Y$.