Homework 3

Due Date: Friday, February 19

1. Let \(\{E_n\}\) be a sequence of Lebesgue measurable subsets of \([0,1]\), and let \[ E \;=\; \{x\in [0,1] \mid x\in E_n\text{ for all but finitely many $n$}\}. \] and \[ F \;=\; \{x\in [0,1] \mid x\in E_n\text{ for infinitely many $n$}\} \]
   1. Prove that \(E\) and \(F\) are Lebesgue measurable.
      Hint: Find sets \(I_1\subseteq I_2\subseteq \cdots\) whose union \(E\), and sets \(J_1 \supseteq J_2\supseteq \cdots\) whose intersection is \(F\).
   2. Prove that if \(m(E) > 0\), then \(\sum_{n\in\mathbb{N}} m(E_n) = \infty\).
   3. Prove that if \(m(E)=m(F)\), then \(m(E_n) \to m(E)\) as \(n\to\infty\).
2. Let \((X,\mathcal{M},\mu)\) be a measure space, and define a function \(\mu^*\colon \mathcal{P}(X) \to [0,\infty]\) by \[ \mu^*(S) \,=\, \inf\{\mu(E) \mid E\in\mathcal{M}\text{ and }S\subseteq E\}. \]
   1. Prove that \(\mu^*\) is an outer measure on \(X\).
   2. Prove that every set in \(\mathcal{M}\) is Carathéodory measurable with respect to \(\mu^*\).
   (Thus every measure \(\mu\) can be obtained by restricting an outer measure.)