Sigma-Algebra Closed under Countable Intersection

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

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

Then $\displaystyle \bigcap_{n \mathop \in \N} E_n \in \Sigma$, where $\displaystyle \bigcap$ denotes set intersection.

Proof
From De Morgan's laws: Complement of Intersection:


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

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


 * $\displaystyle X \setminus \left({X \setminus \left({\bigcap_{n \mathop \in \N} E_n}\right) }\right) = \bigcap_{n \mathop \in \N} E_n$

Combining the previous equalities, it follows that:
 * $\displaystyle \bigcap_{n \mathop \in \N} E_n \in \Sigma$