Sigma-algebra

Let $X$ be a set.

Let $\Sigma$ be a subset of the power set of $X$

Then, $\Sigma$ is a σ-algebra on the set $X$ if the following is true:

1. $X \in \Sigma$ ($X$ is an element of $\Sigma$.)
2. $A \in \Sigma \Rightarrow A^C \in \Sigma$ (For any set, if a set is an element of $\Sigma$, then its complement is in $\Sigma$ also.)
3. $\{A_i\}_{i \in I} \in \Sigma \Rightarrow \bigcup_{i \in I}A_i \in \Sigma$ (If there is a countable collection of sets that are elements of $\Sigma$, then the union of those elements are also in $\Sigma$).

If $\Sigma$ is a σ-algebra on the set $X$, then $(X,\Sigma)$ is a measure space.