\documentclass[12pt]{article}
\input epsf      
\usepackage{graphicx}% Include figure files
\usepackage{amssymb}
\usepackage{amsfonts}
%\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}

\newcommand{\ba}[1]{\begin{array}{#1}}
\newcommand{\ea}{\end{array}}
\newcommand{\arr}[2]{\ba{#1} #2 \ea}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand {\no}  {\noindent}
\newcommand{\borel}{\mathcal{B}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\ra}{\rightarrow}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\half}{\mathbb{H}}
\newcommand{\implies}{\Rightarrow}


\begin{document}

\centerline {\bf Math 523a - Homework 3} 

\bigskip

\begin{enumerate}

\item Problem 18, p. 32 in Folland. 

\item Problem 19, p. 33 in Folland. 

\item Let $\E$ be the collection of open intervals $(a,b)$. 
Define $\rho$ on $E$ by $\rho((a,b))=(b-a)^2$. 
Let $\mu^*$ be the outer measure induced by $\rho$, i.e., 
\bea
\mu^*(E) = \inf \{ \sum_{n=1}^\infty \rho((a_n,b_n)) :
 E \subset \bigcup_{n=1}^\infty (a_n,b_n) \}
\eea
Prove that for every set $E$, $\mu^*(E)=0$. 

\item Let $\E$ be the collection of open intervals $(a,b)$. 
Define $\rho$ on $E$ by $\rho((a,b))=\sqrt{b-a}$. 
Let $\mu^*$ be the outer measure induced by $\rho$.

\begin{enumerate}
\item Compute $\mu^*((a,b))$. (Hint: 
$\sqrt{a+b} \le \sqrt{a}+\sqrt{b}$ for $a,b>0$.)
\item Prove that no open interval is $\mu^*$-measurable. 
\item (optional) Prove that if $E$ is a bounded measurable set, then 
$\mu^*(E)=0$. 
\end{enumerate}

\item We defined the Cantor function in class. It is a continuous
function $F$ on $[0,1]$ with $F(0)=0$, $F(1)=1$ that is constant on 
the complement of the Cantor set. We can extend it to all of $\reals$
by defining it to be $0$ for $x<0$ and $1$ for $x>1$. 
Let $\mu_C$ be the associated Borel measure. Let $m$ be Lebesgue measure. 
\begin{enumerate}
\item Prove that there is a Borel set $S$ such that $m(S)=0$ and 
$\mu_C(E)=\mu_C(S \cap E)$ for all Borel sets $E$. 
\item Prove that there is a Borel set $U$ such that $\mu_C(U)=0$ and 
$m(E)=m(S \cap E)$ for all Borel sets $E$. 
\end{enumerate}

\item Problem 26, p. 39 in Folland. 

\item Problem 30, p. 40 in Folland. 

\item Problem 31, p. 40 in Folland. 

\end{enumerate}



\bigskip
You should do problem 28, p. 39 in Folland, but don't turn it in.  
\end{document}
\end

