Sigma-Algebra Closed under Countable Intersection

Theorem
Let $X$ be a set, and let $\mathcal A$ be a $\sigma$-algebra on $X$.

Suppose that $\left({A_n}\right)_{n \in \N} \in \mathcal A$ is a collection of measurable sets.

Then $\displaystyle \bigcap_{n \in \N} A_n \in \mathcal A$, where $\displaystyle \bigcap$ denotes set intersection.

Proof
Now from De Morgan's laws, it follows that:


 * $\displaystyle \bigcup_{n \in \N} \left({X \setminus A_n}\right) = X \setminus \left({\bigcup_{n \in \N} A_n}\right)$

Also, by Set Difference with Set Difference and Set Union Preserves Subsets:


 * $\displaystyle X \setminus \left({X \setminus \left({\bigcup_{n \in \N} A_n}\right) }\right) = \bigcup_{n \in \N} A_n$

Combining the previous equalities, it follows that$\displaystyle \bigcap_{n \in \N} A_n \in \mathcal A$.