Definition (Product $\sigma$-algebra)

Let $(X,\mathscr{M}),(Y,\mathscr{N})$ be measurable spaces and let $\mathcal{G}$ be the collection of Elementary Sets. We define $$\mathscr{P}=\mathscr{M}\otimes \mathscr{N}=\sigma(\mathcal{G})=\text{"smallest }\sigma\text{-algebra generated by }\mathcal{G}\text{ "}$$where $\mathscr{P}$ is called the product σ-algebra of $\mathscr{M}$ and $\mathscr{N}$.