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 on or 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 $\Sigma \subseteq \mathcal P \left({X}\right)$ (where $\mathcal P \left({X}\right)$ is the power set of $X$) such that:

$(1): \quad X \in \Sigma$

$(2): \quad S \in \Sigma \implies X \setminus S \in \Sigma$

$(3): \quad$ If $\left({S_n}\right)_{n \in \N}$ is a sequence of sets in $\Sigma$, then $\displaystyle \bigcup_{n \mathop \in \N} S_n \in \Sigma$.

Also known as

This is also seen as $\sigma$-algebra, from $\sigma$ being the Greek letter sigma.

Also see

• Results about $\sigma$-algebras can be found here.

Sources

• René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $3.1$