Definition:Product Sigma-Algebra/Binary Case
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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
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.2$