# Definition:Product Measure Space

## Definition

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 $\nu$.