Sigma-algebra

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 $\Sigma$.

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 $X$.