Definition:Product Sigma-Algebra
Definition
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 $S$.
Also see
- Results about product $\sigma$-algebras can be found here.