Definition:Product Sigma-Algebra/Binary Case

Definition
Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces.

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


 * $\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \land S_2 \in \Sigma_2} }$

where:
 * $\sigma$ denotes generated $\sigma$-algebra
 * $\times$ denotes Cartesian product.

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