# Sigma-algebra

A ${\displaystyle \sigma }$-algebra is an algebra that is closed under countable unions. Thus a ${\displaystyle \sigma }$-algebra is a nonempty collection A of subsets of a nonempty set X closed under countable unions and complements. [1]

## ${\displaystyle \sigma }$-algebra Generation

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

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

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

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

## Product ${\displaystyle \sigma }$-algebras

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

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

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

## Other Examples of ${\displaystyle \sigma }$-algebras

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

## Non-examples

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

## References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2
2. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.4