Definition:Product Measure

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

Let $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ be the product measurable space of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$.

The product measure of $\mu$ and $\nu$, denoted $\mu \times \nu$, is the measure defined by:


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

That this uniquely defines a measure on $\Sigma_1 \otimes \Sigma_2$ is shown on Uniqueness of Product Measures and Existence of Product Measures.