Homework 4

Due Date: Friday, February 26

1. Let \(E \subseteq \mathbb{R}\) be a Lebesgue measurable set with \(m(E) < \infty\). Let \(S\subseteq E\), and suppose that \[ m^*(S) + m^*(E-S) \,=\, m(E). \]
   1. Prove that \(m^*(F\cap S)+ m^*(F\cap S^c) = m(F)\) for any Lebesgue measurable set \(F\subseteq E\).
   2. Prove that \(S\) is Lebesgue measurable.
2. Let \(f\colon\mathbb{R} \to\mathbb{R}\). For each \(x\in\mathbb{R}\), the oscillation of \(f\) at \(x\) is defined by \[ \mathrm{osc}_f(x) \,=\, \inf\bigl\{\mathrm{diam}\bigl(f(I)\bigr) \;\bigl|\; I\text{ is an open interval containing }x\bigr\}, \] where \[ \mathrm{diam}(S) \,=\, \sup\bigl\{|x-y| \;\bigl|\; x,y\in S\bigr\} \] for any set \(S\subseteq\mathbb{R}\).
   1. If \(x\in\mathbb{R}\), prove that \(f\) is continuous at \(x\) if and only if \(\mathrm{osc}_f(x) = 0\).
   2. Prove that the set \[ U_b \,=\, \{x\in \mathbb{R} \mid \mathrm{osc}_f(x) < b\} \] is open for every \(b\in(0,\infty)\).
   3. Use parts (a) and (b) to prove that \[ C \,=\, \{x\in \mathbb{R} \mid f\text{ is continuous at }x\}. \] is a \(G_\delta\) set, and hence measurable.
3. Let \(\{f_n\}\) be a sequence of continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).
   1. Prove that the set \[ \{x \in \mathbb{R} \mid f_n(x) \to 0\text{ as }n\to\infty\} \] is a Borel set, and hence measurable.
   2. Prove that the set \[ \bigl\{x \in \mathbb{R} \;\bigl|\; \text{the sequence }\{f_n(x)\}\text{ converges}\} \] is a Borel set, and hence measurable.