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.

$\ds \forall n \in \N: \, $

$\ds E_n$

$\in$

$\ds \Sigma$

$\ds \leadsto \ \ $

$\ds X \setminus E_n$

$\in$

$\ds \Sigma$

$\ds \leadsto \ \ $

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

$\in$

$\ds \Sigma$

$\ds \leadsto \ \ $

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

$\in$

$\ds \Sigma$

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

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

Combining the previous equalities, it follows that:

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

$\blacksquare$