Sigma-Algebra Closed under Countable Intersection

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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.


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

where $\ds \bigcap$ denotes set intersection.


\(\ds \forall n \in \N: \, \) \(\ds E_n\) \(\in\) \(\ds \Sigma\)
\(\ds \leadsto \ \ \) \(\ds X \setminus E_n\) \(\in\) \(\ds \Sigma\) Axiom $(2)$ for $\sigma$-algebras
\(\ds \leadsto \ \ \) \(\ds \bigcup_{n \mathop \in \N} \paren {X \setminus E_n}\) \(\in\) \(\ds \Sigma\) Axiom $(3)$ for $\sigma$-algebras
\(\ds \leadsto \ \ \) \(\ds X \setminus \paren {\bigcup_{n \mathop \in \N} \paren {X \setminus E_n} }\) \(\in\) \(\ds \Sigma\) Axiom $(2)$ for $\sigma$-algebras

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:

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