# Sigma-algebra

A $\sigma$ -algebra is an algebra that is closed under countable unions. Thus a $\sigma$ -algebra is a nonempty collection A of subsets of a nonempty set X closed under countable unions and complements. 

## $\sigma$ -algebra Generation

The intersection of any number of $\sigma$ -algebras on a set $X$ is a $\sigma$ -algebra. The $\sigma$ -algebra generated by a collection of subsets of $X$ is the smallest $\sigma$ -algebra containing $X$ , which is unique by this property.

The $\sigma$ -algebra generated by $E\subseteq 2^{X}$ is denoted as $M(E)$ .

If $E$ and $F$ are subsets of $2^{X}$ and $E\subseteq M(F)$ then $M(E)\subseteq M(F)$ . This result is commonly used to simplify proofs of containment in $\sigma$ -algebras.

An important common example is the Borel $\sigma$ -algebra on $X$ , the $\sigma$ -algebra generated by the open sets of $X$ .

## Product $\sigma$ -algebras

Let $\{X_{\alpha }\}_{\alpha \in A}$ be an indexed collection of nonempty sets, $X=\prod _{\alpha \in A}X_{\alpha }$ , and $\pi _{\alpha }:X\rightarrow X_{\alpha }$ the coordinate maps. If ${\mathcal {M}}_{\alpha }$ is a $\sigma$ -algebra on $X_{\alpha }$ for each $\alpha$ , the product $\sigma$ -algebra on X is generated be $\{\pi _{\alpha }^{-1}(E_{\alpha }):E_{\alpha }\in {\mathcal {M}}_{\alpha },\alpha \in A\}$ . 

Proposition. If A is countable, then $\bigotimes _{\alpha \in A}{\mathcal {M}}_{\alpha }$ is the $\sigma$ -algebra generated by $\{\prod _{\alpha \in A}E_{\alpha }:E_{\alpha }\in {\mathcal {M}}_{\alpha }\}$ .

Proposition. Suppose that ${\mathcal {M}}_{\alpha }$ is generated by ${\mathcal {E}}_{\alpha },\alpha \in A$ . Then $\bigotimes _{\alpha \in A}{\mathcal {M}}_{\alpha }$ is generated by ${\mathcal {F}}_{1}=\{\pi _{\alpha }^{-1}(E_{\alpha }):E_{\alpha }\in {\mathcal {E}}_{\alpha },\alpha \in A\}$ . If A is countable and $X_{\alpha }\in {\mathcal {E}}_{\alpha }$ for all $\alpha$ , then $\bigotimes _{\alpha \in A}{\mathcal {M}}_{\alpha }$ is generated by ${\mathcal {F}}_{2}=\{\prod _{\alpha \in A}E_{\alpha }:E_{\alpha }\in {\mathcal {E}}_{\alpha }\}$ .

## Other Examples of $\sigma$ -algebras

• Given a set $X$ , then $2^{X}$ and $\{\emptyset ,X\}$ are $\sigma$ -algebras, called the indiscrete and discrete $\sigma$ -algebras respectively.
• If $X$ is uncountable, the set of countable and co-countable subsets of $X$ is a $\sigma$ -algebra.
• By Carathéodory's Theorem, if $\mu ^{*}$ is an outer measure on $X$ , the collection of $\mu ^{*}$ -measurable sets is a $\sigma$ -algebra. 
• Let $f:X\to Y$ be a map. If $M$ is a $\sigma$ -algebra on $Y$ , then $\{f^{-1}(E):E\in M\}$ is a $\sigma$ -algebra in $X$ .

## Non-examples

• The algebra of finite and cofinite subsets of a nonempty set $X$ may no longer be a $\sigma$ -algebra. Let $X=\mathbb {Z}$ , then every set of the form $\{2n\}$ for $n\in \mathbb {Z}$ is finite, but their countable union $\bigcup \limits _{n\in \mathbb {Z} }\{2n\}=2\mathbb {Z}$ is neither finite nor cofinite.