Sigma-Algebra Closed under Countable Intersection
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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
\(\ds \forall n \in \N: \, \) | \(\ds E_n\) | \(\in\) | \(\ds \Sigma\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds X \setminus E_n\) | \(\in\) | \(\ds \Sigma\) | 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:
- $\ds \bigcap_{n \mathop \in \N} E_n \in \Sigma$
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.2 \ \text{(iii)}$