91Ó°ÊÓ

Let \(f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto|x| .\) Show that a Borel measurable map \(g:\) \(\mathbb{R} \rightarrow \mathbb{R}\) is \(\sigma(f)=f^{-1}(\mathcal{B}(\mathbb{R}))\)-measurable if and only if \(g\) is even.

Give a counterexample that shows that, in general, the union \(\mathcal{A} \cup \mathcal{A}^{\prime}\) of two \(\sigma\)-algebras need not be a \(\sigma\)-algebra.

Let \(\Omega\) be an uncountably infinite set and \(\mathcal{A}=\sigma(\\{\omega\\}: \omega \in \Omega)\). Show that \(\mathcal{A}=\left\\{A \subset \Omega: A\right.\) is countable or \(A^{c}\) is countable \(\\}\).