Definition:Product of Measurable Spaces/Countable Case

Definition
Let $\sequence {\struct {X_i, \Sigma_i} }_{i \in \N}$ be a sequence of measurable spaces.

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


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

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