Sigma-algebra

This page is under construction.

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.

Examples of $\sigma$-algebras

• Given a set $X$, then $2^{X}$ and $\{\emptyset ,X\}$ are $\sigma$-algebras.

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.