\begin{definition}[Definition:Product Measure Space]
Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces.
The '''product of $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$''' is the measure space:
:$\left({X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}\right)$
where $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ is the product measurable space of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ and $\mu \times \nu$ is the product measure of $\mu$ and $\nu$.
\end{definition}