Sigma Algebra

algebra probability-theory

Definition

Sigma Algebra

Let $\Omega$ be a non-empty set. A $\sigma$-algebra on $\Omega$ is a collection of subsets $\mathfrak{S}\subseteq \mathcal{P}(\Omega )$ such that:

1. $\Omega \in \mathfrak{S}$
2. If $A\in \mathfrak{S}$, then $\Omega \setminus A\in \mathfrak{S}$
3. If $A_1,A_2,A_3,\cdots \in \mathfrak{S}$, then ${\bigcup }_{i=1}^{\infty }{A}_i\in \mathfrak{S}$
4. If $A_1,A_2,A_3,\cdots \in \mathfrak{S}$, then ${\bigcap }_{i=1}^{\infty }{A}_i\in \mathfrak{S}$

A pair $(\Omega ,\mathfrak{S})$ is called a measurable space.