Definition:Sigma-Algebra

Definition
A sigma-algebra is a sigma-ring with a unit.

Thus, a sigma-algebra is an algebra of sets which is closed under countable unions.

Formal Definition
Let $$X$$ be a set.

A sigma-algebra, or $$\sigma$$-algebra, over$$X$$ is a non-empty 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 $$X$$ is a set $$\mathcal A \subseteq \mathcal P \left({X}\right)$$ (where $$\mathcal P \left({X}\right)$$ is the power set of $$X$$) such that:


 * 1) $$\mathcal A \ne \varnothing$$;
 * 2) $$S \in \mathcal A \implies X \setminus S \in \mathcal A$$;
 * 3) If $$\left\{{S_i}\right\}_{i \in \N}$$ is a countable collection of sets in $$\mathcal A$$, then $$\bigcup_{i \in \N} S_i \in \mathcal A$$.