Existence and Uniqueness of the Product Measure

Let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be measure spaces, and suppose that both $\mu$ and $\nu$ are $\sigma$-finite. Then there exists exactly one measure $\mu\otimes\nu$ on the product $\sigma$-algebra $\mathcal{F}\otimes\mathcal{G}$ such that

for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$, with the product in $[0,\infty]$ understood with the conventions of Measure, Measure Space, and Probability Measure. The measure $\mu\otimes\nu$ is itself $\sigma$-finite and is called the product measure. Uniqueness rests on Dynkin's Pi-Lambda Theorem applied to the $\pi$-system of measurable rectangles.