Measurable Rectangles and the Product σ-Algebra

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces, and let $X\times Y$ be their Cartesian product.

(i) Measurable rectangles. A measurable rectangle is a set $A\times B$ with $A\in\mathcal{F}$ and $B\in\mathcal{G}$. The set of measurable rectangles is

$$\mathrm{Rect}(\mathcal{F},\mathcal{G})=\{A\times B:A\in\mathcal{F},\ B\in\mathcal{G}\},$$

a set of subsets of $X\times Y$ (Sets and Maps: Ordinary Notation (vii)).

(ii) Product $\sigma$-algebra. The product $\sigma$-algebra of $\mathcal{F}$ and $\mathcal{G}$ is the $\sigma$-algebra on $X\times Y$ generated by the measurable rectangles,

$$\mathcal{F}\otimes\mathcal{G}=\sigma\bigl(\mathrm{Rect}(\mathcal{F},\mathcal{G})\bigr).$$

It is a $\sigma$-algebra on $X\times Y$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), so $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ is a measurable space.