Sigma Algebra

Statements

Definition. (Citation Needed) Let $\Omega$ be some set, and let $P^{\ast }(\Omega )$ represent its power set. Then a subset $\mathcal{F}\subseteq P^{\ast }(\Omega )$ is called a $\sigma$-algebra if it satisfies the following three properties:

1. $\Omega$ is in $\mathcal{F}$, and $\Omega$ is considered to be the universal set in the following context.

2. $\mathcal{F}$ is closed under complementation:

   $$A\subset \mathcal{F}\Rightarrow \Omega \setminus A.$$

3. $\mathcal{F}$ is closed under countable unions:

   $$\{A_i{\}}_{i\in \mathbb{N}}\Rightarrow \bigcup _{i\in \mathbb{N}}{A}_i.$$

References

σ-algebra - Wikipedia