# Category:Definitions/Product Sigma-Algebras

This category contains definitions related to Product Sigma-Algebras.
Related results can be found in Category:Product Sigma-Algebras.

### Binary Case

Let $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ be measurable spaces.

The product $\sigma$-algebra of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as:

$\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \text { and } S_2 \in \Sigma_2} }$

where:

$\sigma$ denotes generated $\sigma$-algebra
$\times$ denotes Cartesian product.

This is a $\sigma$-algebra on the Cartesian product $X \times Y$.

### Finite Case

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 $S$.

### Countable Case

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

Let:

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

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 $S$.

## Pages in category "Definitions/Product Sigma-Algebras"

The following 4 pages are in this category, out of 4 total.