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 $\struct {X_1, \Sigma_1}, \struct {X_2, \Sigma_2}, \ldots$ 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 $\Sigma_1, \Sigma_2, \ldots$.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $8.1$: Polish Spaces