Measurable space

If $\mathrm{\Omega }$ is a given set, then a $\sigma$-algebra $\mathcal{F}$ on $\mathrm{\Omega }$ is a family $\mathcal{F}$ of subsets (measurable sets) of $\mathrm{\Omega }$ with the following properties:

1. Empty Set:
   $\mathrm{\varnothing }\in \mathcal{F}$
2. Closed under Complements:
   $F\in \mathcal{F}\Rightarrow F^C\in \mathcal{F}$, where $F^C=\mathrm{\Omega }\setminus F$ is the complement of $F$ in $\mathrm{\Omega }$
3. Closed under Countable Unions:
   $A_1,A_2,\dots \in \mathcal{F}\Rightarrow A:=\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in \mathcal{F}$

The pair $(\mathrm{\Omega },\mathcal{F})$ is called a measurable space.

Here we can define a measure and obtain a measure space.