Definition:Product of Measurable Spaces/Finite Case

Definition
Let $n \in \N$.

Let $\struct {X_1, \Sigma_1}, \struct {X_2, \Sigma_2}, \ldots, \struct {X_n, \Sigma_n}$ be measurable spaces.

The product of $\struct {X_1, \Sigma_1}, \struct {X_2, \Sigma_2}, \ldots, \struct {X_n, \Sigma_n}$ is the measurable space:


 * $\ds \struct {\prod_{i \mathop = 1}^n X_i, \bigotimes_{i \mathop = 1}^n \Sigma_i}$

where $\ds \bigotimes_{i \mathop = 1}^n \Sigma_i$ denotes the product $\sigma$-algebra of $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$.