Sigma-algebra - Imported Knowlpedia Content

A sigma-algebra on a set $X$ is a nonempty collection $\Sigma \subseteq \mathcal P(X)$ such that if $A\in\Sigma$ then $X\setminus A\in\Sigma$ , and whenever $(A_n)_{n\ge 1}$ is a sequence in $\Sigma$ , the union $\bigcup_{n=1}^\infty A_n$ lies in $\Sigma$ .

Equivalent characterizations

Equivalently, $\Sigma$ contains $X$ and is closed under complements and countable unions; it then automatically contains the empty set and is closed under countable intersections. Sigma-algebras define measurable spaces and are the domains of measures.

Examples

For any $X$ , the power set $\mathcal P(X)$ is a sigma-algebra.
On $\mathbb R$ with its usual topology, the Borel sigma-algebra is a sigma-algebra.
The “trivial” sigma-algebra $\{\varnothing, X\}$ is a sigma-algebra on any set $X$ .