Existence of Product Measures

Theorem
Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces.

Then there exists a $\sigma$-finite measure $\rho$ on the product space $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ such that:


 * $\forall E_1 \in \Sigma_1, E_2 \in \Sigma_2: \rho \left({E_1 \times E_2}\right) = \mu \left({E_1}\right) \nu \left({E_2}\right)$

This $\rho$ is called product measure and is also denoted $\mu \times \nu$.