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:


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