Definition:Product of Measurable Spaces

From ProofWiki
Jump to navigation Jump to search

Definition

Binary Case

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

The product of $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ is the measurable space:

$\struct {X_1 \times X_2, \Sigma_1 \otimes \Sigma_2}$

where $\Sigma_1 \otimes \Sigma_2$ denotes the product $\sigma$-algebra of $\Sigma_1$ and $\Sigma_2$.


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.


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 $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$.


Countable Case

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

The product of $\struct {X_1, \Sigma_1}, \struct {X_2, \Sigma_2}, \ldots$ is the measurable space:

$\ds \struct {\prod_{i \mathop = 1}^\infty X_i, \bigotimes_{i \mathop = 1}^\infty \Sigma_i}$

where $\ds \bigotimes_{i \mathop = 1}^\infty \Sigma_i$ denotes the product $\sigma$-algebra of $\Sigma_1, \Sigma_2, \ldots$.

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Product_of_Measurable_Spaces&oldid=554308"