Definition:Product Sigma-Algebra/Countable Case

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

Let:


 * $\ds S = \set {\prod_{i \mathop = 1}^n A_i \times \prod_{i \mathop = n + 1}^\infty \Omega : n \in \N, \, A_i \in \Sigma_i \text { for each } 1 \le i \le n}$

We define the product $\sigma$-algebra of $\Sigma_1, \Sigma_2, \ldots$, written $\ds \bigotimes_{i \mathop = 1}^\infty \Sigma_i$, by:


 * $\ds \bigotimes_{i \mathop = 1}^\infty \Sigma_i = \map \sigma S$

where $\map \sigma S$ denotes the $\sigma$-algebra generated by $S$.