Sigma Algebra

algebra probability-theory

Definition

Sigma Algebra

Let $\Omega$ be a non-empty set. A collection of subsets of $\Omega$, denoted by $\mathfrak{S}$, is a $\sigma$-algebra if it satisfies the following properties:

1. Inclusion of $\Omega$: $\Omega \in \mathfrak{S}$
2. Closure under complement: $\forall A\in \mathfrak{S}:(A^c=\Omega \setminus A)\in \mathfrak{S}$
3. Closure under countable unions: $\forall A_1,A_2,A_3,\cdots \in \mathfrak{S}:({\bigcup }_{i=1}^{\infty }{A}_i)\in \mathfrak{S}$.
4. Closure under countable intersections: $\forall A_1,A_2,A_3,\cdots \in \mathfrak{S}:({\bigcap }_{i=1}^{\infty }{A}_i)\in \mathfrak{S}$.