Definition:Product of Measurable Spaces/Binary Case

Definition
Let $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ be measurable spaces.

The product of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ is the measurable space:


 * $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$

where $\Sigma_1 \otimes \Sigma_2$ denotes the product $\sigma$-algebra of $\Sigma_1$ and $\Sigma_2$.