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

If $A$ is a countable set, then $\bigotimes \limits _{\alpha \in A}M_{\alpha }$ is the $\sigma$-algebra generated by $\left\{\prod \limits _{\alpha \in A}E_{\alpha }:E_{\alpha }\in M_{\alpha }\right\}$. This is called the product $\sigma$-algebra.

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.