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\title{Real Analysis}
\author{Midterm Exam}
\date{October 22, 2008}
\maketitle

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\begin{enumerate}

\item Let $(\Omega, \Sigma, \mu)$ be a measure space and $E \in
  \Sigma$.  Show that
\[
  \mu_E(A)  = \mu(A \cap E)  
  \qquad \text{for all $A \in \Sigma$}
\]
defines a measure.
\hfill (10)


\item Show that a monotonic function on $\R$ is Borel-measurable.
\hfill (10)


\item Let $(\Omega, \Sigma, \mu)$ be a measure space and $f \colon
  \Omega \to [0,\infty]$ a measurable function.  Show that, for
  $0<p<\infty$, 
\[
  \int_\Omega f^p \, \d \mu
  = p \int_0^\infty t^{p-1} \, \mu (\{x \in \Omega \colon f(x)>t \})
  \, \d t \,.
\]
\hfill (10)


\item \emph{Theorem on differentiation under the integral.}

Let $(\Omega, \Sigma, \mu)$ be a measure space and let $I
  \subset \R$ open.  Suppose that $f \colon \Omega \times I \to \R$
  satisfies
\begin{itemize}
\item[(i)] $f( \cdot, t)$ is measurable for every fixed $t \in I$;
\item[(ii)] $f( x, \cdot)$ is differentiable for almost every fixed $x
  \in \Omega$;
\item[(iii)] $\lvert \partial f(x,t)/\partial t \rvert \leq g(x)$ for some $g
  \in L^1(\Omega)$.
\end{itemize}
Then
\[
  \frac{\d}{\d t} \int_\Omega f(x,t) \, \d \mu(x)
  = \int_\Omega \frac{\partial f(x,t)}{\partial t} \, \d \mu(x) 
\]
for every $t \in I$.

\begin{enumerate}
\item Prove the theorem.
\item Let
\[
  I(t) = \int_0^\infty \frac{\sin x}x \, \e^{-tx} \, \d x \,.
\]
Verify that you may differentiate under the integral, then compute
$I'(t)$. 
\item Use the result from (b) to show that the improper integral
\[
  \int_0^\infty \frac{\sin x}x \, \d x = \arctan (\infty) = \frac\pi2
  \,.
\]

% \item  Provide an alternative justification of this computation
% which relies on Fubini's theorem?
\end{enumerate}
\hfill (10+10+10)


\item Let $(\Omega, \Sigma, \mu)$ be a finite measure space.  

Show
that for $1 \leq p \leq r \leq \infty$, $L^p(\Omega) \supset
L^r(\Omega)$. 
\hfill (10)


\item Let $(\Omega, \Sigma, \mu)$ be a measure space, $1 < p \leq
\infty$ and $1/p+1/q=1$.  Let $\ell \in L^p(\Omega)^*$ be represented
by a function $g \in L^q(\Omega)$ in the sense that
\[
  \ell(f) = \int_\Omega f \, g \, \d \mu 
\]
for every $f \in L^p(\Omega)$.  Show that
\[
  \lVert \ell \rVert_{L^p(\Omega)^*}
  = \lVert g \rVert_{L^q(\Omega)} \,.
\]
\hfill (10)


\item 
\begin{enumerate}
\item Find a sequence of bounded, Lebesgue-measurable sets in $\R$
whose characteristic functions converge weakly in $L^2(\R)$ to a
function $f$ with the property that $2f$ is a characteristic function
of a set with positive measure.

\emph{Note:} Full credit if you verify that your example has the
requested properties on some (nontrivial) subspace of $L^2$.

\item How about the possibility that $f/2$ is a characteristic
function?
\end{enumerate}
\hfill (10+10)
\end{enumerate}


\end{document}
