Sigma-Algebra

Definition

A sigma-algebra $\mathcal{F}$ is a Family of Sets satisfying the following properties

• $\Omega \in \mathcal{F}\text{or}\mathcal{F}\ne \varnothing$
• $\forall A\in \mathcal{F},A^{\complement }\in \mathcal{F}$
• $\forall \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{F},\bigcup _{i=1}^{\infty }{A}_i\in \mathcal{F}$

where $\Omega$ is a universal set

Intersection of Sigma-Fields

Consider a set of sigma-fields ${\mathcal{F}}_i$ on $\Omega$,

• ${\mathcal{F}}_1\cap {\mathcal{F}}_2$ is a sigma-field on $\Omega$.
• $\bigcup _{i=1}^{\infty }{\mathcal{F}}_i$ is a sigma-field on $\Omega$.
• $\bigcup ^{\infty }{\mathcal{F}}_i$ is a sigma-field on $\Omega$

Facts

If a set of subsets $\mathcal{A}$ of a universal set $\Omega$ is both Pi-System and Lambda-System, then $\mathcal{A}$ is a sigma-algebra on $\Omega$.