Definition:Sigma-Algebra

A sigma-algebra, or $$\sigma$$-algebra, over a set $$X$$ is a collection of subsets of $$X$$ which is closed under complements and countable unions, and includes the set $$X$$ itself.

More formally, a $$\sigma$$-algebra over a set $$X$$ is a set $$\mathcal{A} \subseteq \mathcal{P}(X)$$ (where $$\mathcal{P}(X)$$ is the power set of $$X$$) such that:


 * 1) If $$S \in \mathcal{A}$$, then $$X \setminus S \in \mathcal{A}$$.
 * 2) If $$\{S_i\}_{i \in \N}$$ is a countable collection of sets in $$\mathcal{A}$$, then $$\cup_{i \in \N} S_i \in \mathcal{A}$$.