\begin{definition}[Definition:Product Sigma-Algebra/Binary Case]
Let $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ be measurable spaces.
The '''product $\sigma$-algebra''' of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as:
:$\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \text { and } S_2 \in \Sigma_2} }$
where:
:$\sigma$ denotes generated $\sigma$-algebra
:$\times$ denotes Cartesian product.
This is a $\sigma$-algebra on the Cartesian product $X \times Y$.
\end{definition}