Let $X$ be a set. A \textbf{$\sigma$-algebra} on $X$ is a \reftext{def:family-subfamily-subsets-set-2026a}{family} $\mathcal{F}$ of subsets of $X$ with the following three properties.

1. $X\in\mathcal{F}$.
2. If $A\in\mathcal{F}$, then the \reftext{def:complement-subset-relative-set-2026a}{complement} $X\setminus A$ belongs to $\mathcal{F}$.
3. For every \reftext{def:sequence-in-set-2026a}{sequence} $(A_m)_{m\in\mathbb{N}}$ of members of $\mathcal{F}$, indexed by the \reftext{def:natural-numbers-2026a}{natural numbers}, the union $\bigcup_{m\in\mathbb{N}}A_m$ belongs to $\mathcal{F}$.

The pair $(X,\mathcal{F})$ is called a \textbf{measurable space}, and the members of $\mathcal{F}$ are called \textbf{measurable sets}.

It follows from properties 1 and 2 that $\varnothing\in\mathcal{F}$, and from properties 2 and 3 that $\mathcal{F}$ is closed under countable intersections, since $\bigcap_{m}A_m=X\setminus\bigcup_{m}(X\setminus A_m)$; taking sequences with finitely many distinct terms, $\mathcal{F}$ is also closed under finite unions and finite intersections.