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\begin{document}

\title{Real Analysis}
\author{Homework 5}
\date{due October 8, 2008}
\maketitle

\begin{enumerate}

\item (From Folland.) Let
\[
  f(x,y) = (1-xy)^{-a}
\]
with $a>0$.  Investigate the existence and equality of
\[
  \int_{[0,1]\times[0,1]} f \, \d (x,y) \,, \qquad
  \int_0^1 \int_0^1 f(x,y) \, \d x \, \d y \,, \qquad \text{and} \qquad
  \int_0^1 \int_0^1 f(x,y) \, \d y \, \d x \,.
\]

\item (From Folland.) Show that
\[  
  \int_0^\infty \e^{-sx} \, x^{-1} \, \sin^2x \, \d x 
  = \frac14 \, \ln(1+4 s^{-2})
\]
for $s>0$ by integrating $\e^{-sx} \, \sin(2xy)$ with respect to $x$
and $y$.

\item (From Lieb and Loss.) Let $(\Omega, \Sigma, \mu)$ be a finite measure
space, let $f$ and $\{f_j\}_{j \in \N}$ be complex-valued, measurable
functions on $\Omega$ with
\[
  \lim_{j \to \infty} f_j(x) = f(x)
\]
for almost every $x \in \Omega$, and assume that
\[
  \int_\Omega \lvert f_j \rvert^2 \, \d \mu < 1
\]
and
\[
  \int_\Omega \lvert f \rvert^2 \, \d \mu < \infty \,.
\]
\begin{enumerate}
\item Prove that 
\[
  \int_\Omega \lvert f_j - f \rvert^p \, \d \mu \to 0
\]
as $j \to \infty$ for any $0<p<2$.  
\item Construct a counter-example to show that this can fail for
  $p=2$.

\end{enumerate}
\item Let $(\Omega, \Sigma, \mu)$ be a measure space and $f \in
  L^\infty(\Omega) \cap L^q(\Omega)$ for some $q \in [1,\infty)$.
  Then $f \in L^p(\Omega)$ for all $p>q$ and
\[
  \lVert f \rVert_\infty = \lim_{p \to \infty} \lVert f \rVert_p \,.
\]
\end{enumerate}

\end{document}
