# 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

 $\ds \forall n \in \N: \ \$ $\ds E_n \in \Sigma$ $\implies$ $\ds X \setminus E_n \in \Sigma$ Axiom $(2)$ for $\sigma$-algebras $\ds$ $\implies$ $\ds \bigcup_{n \mathop \in \N} \left({X \setminus E_n}\right) \in \Sigma$ Axiom $(3)$ for $\sigma$-algebras $\ds$ $\implies$ $\ds X \setminus \left({\bigcup_{n \mathop \in \N} \left({X \setminus E_n}\right) }\right) \in \Sigma$ Axiom $(2)$ for $\sigma$-algebras
$\displaystyle \bigcup_{n \mathop \in \N} \left({X \setminus E_n}\right) = X \setminus \left({\bigcap_{n \mathop \in \N} E_n}\right)$
$\displaystyle X \setminus \left({X \setminus \left({\bigcap_{n \mathop \in \N} E_n}\right) }\right) = \bigcap_{n \mathop \in \N} E_n$

Combining the previous equalities, it follows that:

$\displaystyle \bigcap_{n \mathop \in \N} E_n \in \Sigma$

$\blacksquare$