Product Sigma-Algebra

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces. A measurable rectangle is a subset of the Cartesian product $X\times Y$ of the form $A\times B$ with $A\in\mathcal{F}$ and $B\in\mathcal{G}$.

The product $\sigma$-algebra $\mathcal{F}\otimes\mathcal{G}$ on $X\times Y$ is the $\sigma$-algebra generated by the family of all measurable rectangles.