Alternative characterization of \(\mathscr{B}\left(\mathbb{R}^{n}ight)\). In older books the Borel sets are often introduced as the smallest family \(\mathscr{M}\) of sets which is stable under countable intersections of decreasing and countable unions of increasing sequences of sets, and which contains all open sets \(\mathscr{O}\). Use Problem 3.14 to show that \(\mathscr{M}=\mathscr{B}\left(\mathbb{R}^{n}ight)\). Can we omit 'decreasing' and 'increasing' in the above characterization, i.e. is \(\mathscr{B}\left(\mathbb{R}^{n}ight)\) also the smallest family containing countable intersections and countable unions of its members, and all open sets?

Data from problem 3.14

 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.