Definition:Product of Measurable Spaces/Finite Case
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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.
The product of $\struct {X_1, \Sigma_1}, \struct {X_2, \Sigma_2}, \ldots, \struct {X_n, \Sigma_n}$ is the measurable space:
- $\ds \struct {\prod_{i \mathop = 1}^n X_i, \bigotimes_{i \mathop = 1}^n \Sigma_i}$
where $\ds \bigotimes_{i \mathop = 1}^n \Sigma_i$ denotes the product $\sigma$-algebra of $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$.