Let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be \reftext{def:measure-measure-space-2026a}{measure spaces}, and suppose that both $\mu$ and $\nu$ are $\sigma$-finite. Then there exists exactly one measure $\mu\otimes\nu$ on the \reftext{def:product-sigma-algebra-2026a}{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 \ref{def:measure-measure-space-2026a}. The measure $\mu\otimes\nu$ is itself $\sigma$-finite and is called the \textbf{product measure}. Uniqueness rests on \ref{lem:dynkin-pi-lambda-2026a} applied to the $\pi$-system of measurable rectangles.