Sigma-Algebra and Measurable Space

Let $X$ be a set. A $\sigma$-algebra on $X$ is a 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 complement $X\setminus A$ belongs to $\mathcal{F}$.
3. For every sequence $(A_m)_{m\in\mathbb{N}}$ of members of $\mathcal{F}$, indexed by the natural numbers, the union $\bigcup_{m\in\mathbb{N}}A_m$ belongs to $\mathcal{F}$.

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