Monotone classes (2). Recall from Problem 3.14 that a monotone class \(\mathscr{M} \subset \mathscr{P}(X)\) is a family which contains \(X\) and is stable under countable unions of increasing sets and countable intersections of decreasing sets; denote by \(\mathfrak{m}(\mathscr{G})\) the smallest monotone class containing the family \(\mathscr{G} \subset \mathscr{P}(X)\). Show the following analogues of Lemma 5.4 and Theorem 5.5 .

(i) \(\mathscr{M}\) is a \(\sigma\)-algebra if, and only if, it is stable under the formation of complements.

(ii) If \(\mathscr{G}\) is stable under the formation of complements and finite intersections, then \(\mathscr{M}=\) \(\mathfrak{m}(\mathscr{G})\) is a \(\sigma\)-algebra (namely \(\sigma(\mathscr{G})\) ).

[use the monotone class theorem from Problem 3.14 .]

Data from problem 3.14

Monotone classes (1). A family \(\mathscr{M} \subset \mathscr{P}(X)\) which contains \(X\) and is stable under countable unions of increasing sets and countable intersections of decreasing sets
\[
\begin{array}{ll}
\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{M}, & A_{1} \subset \cdots \subset A_{n} \subset A_{n+1} \uparrow A=\bigcup_{n \in \mathbb{N}} A_{n} \Longrightarrow A \in \mathscr{M} \\
\left(B_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{M}, & B_{1} \supset \cdots \supset B_{n} \supset B_{n+1} \downarrow B=\bigcap_{n \in \mathbb{N}} B_{n} \Longrightarrow B \in \mathscr{M}
\end{array}
\]
is called a monotone class. Assume that \(\mathscr{M}\) is a monotone class and \(\mathscr{F} \subset \mathscr{P}(X)\) any family of sets.
(i) Mimic the proof of Theorem 3.4 to show that there is a minimal monotone class \(\mathfrak{m}(\mathscr{F})\) such that \(\mathscr{F} \subset \mathfrak{m}(\mathscr{F})\).
(ii) If \(\mathscr{F}\) is stable w.r.t. complements, i.e. \(F \in \mathscr{F} \Longrightarrow F^{c} \in \mathscr{F}\), then so is \(\mathfrak{m}(\mathscr{F})\).
(iii) If \(\mathscr{F}\) is \(\cap\)-stable, i.e. \(F, G \in \mathscr{F} \Longrightarrow F \cap G \in \mathscr{F}\), then so is \(\mathfrak{m}(\mathscr{F})\).
[ show that the families
\[
\begin{aligned}
\Sigma & :=\{M \in \mathfrak{m}(\mathscr{F}): M \cap F \in \mathfrak{m}(\mathscr{F}) \forall F \in \mathscr{F}\} \\
\Sigma^{\prime} & :=\{M \in \mathfrak{m}(\mathscr{F}): M \cap N \in \mathfrak{m}(\mathscr{F}) \forall N \in \mathfrak{m}(\mathscr{F})\}
\end{aligned}
\]
are again monotone classes satisfying \(\mathscr{F} \subset \Sigma, \Sigma^{\prime}\).]
(iv) Use (i)-(iii) to prove the following.
Monotone class theorem. Let \(\mathscr{F} \subset \mathscr{P}(X)\) be a family of sets which is stable under the formation of complements and intersections. If \(\mathscr{M} \supset \mathscr{F}\) is a monotone class, then \(\mathscr{M} \supset \sigma(\mathscr{F})\).

Data from lemma 5.4

Data from theorem 5.5

A Dynkin system D is a o-algebra if, and only if, it is stable under finite intersections: D, E DnEED. Proof Since a o-algebra is n-stable (see Properties 3.2 and Problem 3.1) as well as a Dynkin system (Remark 5.2) it remains only to show that a n-stable Dynkin system is a o-algebra. Let (Dn)neN be a sequence of sets in 9. Let us show that D:=UNEN D ED. Set E=D and En+1=(((Dn+1 \ Dn) \ Dn-1) \) \ D =D+1DD-1... DiED where we use (D) and the assumed n-stability of 9. The En are obviously mutually disjoint and D= UNENE E by (D3). Lemma 5.4 is not applicable if is given in terms of a generator, which is often the case. The next theorem is very important for applications as it extends Lemma 5.4 to the much more convenient setting of generators.