Math 461 Bard College

Homework 3

Due Date: Friday, February 19

  1. Let \(\{E_n\}\) be a sequence of Lebesgue measurable subsets of \([0,1]\), and let \[ E \;=\; \{x\in [0,1] \mid x\in E_n\text{ for infinitely many $n$}\} \] and \[ F \;=\; \{x\in [0,1] \mid x\in E_n\text{ for all but finitely many $n$}\}. \]
    1. Prove that \(E\) and \(F\) are Lebesgue measurable.
    2. Prove that if \(m(F) > 0\), then \(\sum_{n=1}^\infty m(E_n) = \infty\).
    3. Prove that if \(E=F\), 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.)
  3. 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.