Sigma-Algebra Closed under Countable Intersection

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

Suppose that $\sequence {E_n}_{n \mathop \in \N} \in \Sigma$ is a collection of measurable sets.

Then:
 * $\ds \bigcap_{n \mathop \in \N} E_n \in \Sigma$,

where $\ds \bigcap$ denotes set intersection.

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


 * $\ds \bigcup_{n \mathop \in \N} \paren {X \setminus E_n} = X \setminus \paren {\bigcap_{n \mathop \in \N} E_n}$

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


 * $\ds X \setminus \paren {X \setminus \paren {\bigcap_{n \mathop \in \N} E_n} } = \bigcap_{n \mathop \in \N} E_n$

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