$-algebra on $X$

Let $X$ be a set and let $\mathcal{P}(X)$ denote the power set of $X$. A $\sigma$-algebra $\mathcal{A} \subseteq \mathcal{P}(X)$ is a collection of subsets of $X$ that satisfies the following conditions:

1. $\emptyset \in \mathcal{A}$.
2. If $A \in \mathcal{A}$, then its complement, $A^c = X\setminus A$, is also in $\mathcal{A}$.
3. If $(A_n)_{n=1}^{\infty}$ is a sequence of sets in $\mathcal{A}$, then their union, $\bigcup_{n=1}^{\infty}{A_n}\in\mathcal{A}$.

The elements of a $\sigma$-algebra are called measurable sets.