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.