Product Measure

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be measure spaces with $\mu$ and $\nu$ $\sigma$-finite, and let $\mathcal{F}\otimes\mathcal{G}$ be the product $\sigma$-algebra on $X\times Y$.

(i) Product measure. The product measure $\mu\otimes\nu$ is the measure on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$ with

$$(\mu\otimes\nu)(A\times B)=\mu(A)\,\nu(B)\qquad\text{for all }A\in\mathcal{F}\text{ and }B\in\mathcal{G},$$

which exists and is unique by Existence and Uniqueness of the Product Measure (i). The measure space $(X\times Y,\mathcal{F}\otimes\mathcal{G},\mu\otimes\nu)$ is the product measure space.