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.