Definition:Product Sigma-Algebra/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.

Let:

$\ds S = \set {\prod_{i \mathop = 1}^n S_i : S_i \in \Sigma_i \text { for each } i \in \set {1, 2, \ldots, n} }$

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

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

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