Definition:Product Sigma-Algebra/Binary Case

Definition

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 $X \times Y$.

Also see

• Definition:Product of Measurable Spaces: Binary Case

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.2$