Sigma-Ring is Closed under Countable Intersections
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Theorem
Let $\RR$ be a $\sigma$-ring.
Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a sequence of sets in $\RR$.
Then:
- $\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$
Proof
\(\ds \forall n \in \N_{>0}: \, \) | \(\ds A_1, A_n \in \RR\) | \(\leadsto\) | \(\ds A_1 \setminus A_n \in \RR\) | Axiom $(\text {SR} 2)$ for $\sigma$-rings | ||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} \in \RR\) | Axiom $(\text {SR} 3)$ for $\sigma$-rings | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds A_1 \setminus \paren {\bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} } \in \RR\) | Axiom $(\text {SR} 2)$ for $\sigma$-rings |
From De Morgan's laws: Difference with Intersection:
- $\ds \bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} = A_1 \setminus \paren {\bigcap_{n \mathop = 2}^\infty A_n}$
From Set Difference with Set Difference:
\(\ds A_1 \setminus \paren {A_1 \setminus \paren {\ds \bigcap_{n \mathop = 2}^\infty A_n} }\) | \(=\) | \(\ds A_1 \cap \paren {\ds \bigcap_{n \mathop = 2}^\infty A_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {\bigcap_{n \mathop = 1}^\infty A_n}\) |
Combining the previous equalities, it follows that:
- $\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$
$\blacksquare$
Sources
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras: Problem $1.1.3$