\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage}

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\Ei{\operatorname{Ei}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\Q{{\mathbb Q}}

\begin{document}

\title{Real Analysis}
\author{Homework 3}
\date{due September 24, 2008}
\maketitle

\begin{enumerate}

\item Let $\mu$ denote the Lebesgue measure on $\R$.  Give a
  self-contained proof of the \emph{outer regularity} of $\mu$, i.e.,
  show that
\[
  \mu(A) = \inf \{ \mu(E) \colon E \text { open and } A \subset E \}
  \,. 
\]

\item (From Lieb and Loss.) 
\begin{enumerate}
\item Show that $F \colon \Omega \to \R$ is measurable if and only if
$\{ x \in \Omega \colon f(x)>a \}$ is measurable for every $a \in
\Q$. 
\item For $a \in \Q$ and $f,g$ measurable, show that
\[
  \{x \in \Omega \colon f(x)+g(x)>a \}
  = \bigcup_{b \text{ rational}}
  \{x \in \Omega \colon f(x)>b \} \cap
  \{x \in \Omega \colon g (x)>a-b \} \,.
\]
Conclude that $f+g$ measurable.

\item Similarly, show that $fg$ measurable if $f$ and $g$ are measurable.
\end{enumerate}

\item (From Lieb and Loss.)  Let $f \colon \R^n \to \R$ be defined by
\[
  f(x) = \lvert x \rvert^{-p} \, \chi_{\{\lvert x \rvert < 1 \}}(x)
  \,.
\]
Compute the integral of $f$ over $\R^n$ in two ways:
\begin{enumerate}
\item Use polar coordinates and compute the integral by the standard
  rules of calculus.

\item Compute $\mathcal L^n(\{ x \in \R^n \colon f(x)>a\})$ and use
  the definition of the Lebesgue integral.
\end{enumerate}

\item Prove that a function $f \colon \R^n \to \R$ is continuous if
  and only if it is \emph{upper semicontinuous} (the set $\{ x \colon
  f(x)<t \}$ is open for all $t$) and \emph{lower semicontinuous} (the
  set $\{ x \colon f(x)>t \}$ is open for all $t$).

\end{enumerate}

\end{document}
