sigma algebra

A $\sigma$-algebra is a set of subsets of a space such that it is are closed under set union and set complement.

Formally,
let $X$ be a space, and $\mathcal{P}(X)$ be its power set. A subset $\mathrm{\Sigma }\subseteq \mathcal{P}(X)$ is called a $\sigma$-algebra if

1. $X\in \mathrm{\Sigma }$
2. $(A_1\cup A_2\cup \cdots )\in \mathrm{\Sigma }$ for all $A_i\in \mathrm{\Sigma }$
3. $(X\setminus A)\in \mathrm{\Sigma }$ for all $A\in \mathrm{\Sigma }$

By De Morgan's laws, $(A_1\cap A_2\cap \cdots )\in \mathrm{\Sigma }$ for all $A_i\in \mathrm{\Sigma }$

By (1) and (3), $\varnothing \in \mathrm{\Sigma }$.

An element of a $\sigma$-algebra is called a measurable set.
An ordered pair $(X,\mathrm{\Sigma })$ is called a measurable space.
A tuple $(X,\mathrm{\Sigma },\mu )$ of a measurable space with measure $\mu$ is a measure space.