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\begin{document}

\title{Real Analysis}
\author{Homework 4}
\date{due October 1, 2008}
\maketitle

\begin{enumerate}

\item (From Lieb and Loss.) Write out a simple sufficient condition on
  $f_n$ such that
\[
  \sum_{n=0}^\infty \int_\Omega f_n(x) \, \d \mu(x)
  = \int_\Omega \sum_{n=0}^\infty f_n(x) \, \d \mu(x) \,.
\]

\item Let $(\Omega_i, \Sigma_i, \mu_i)$ for $i=1,2,3$ be
  $\sigma$-finite measure spaces.  Show that the product measure is
  associative, i.e.
\[
  \mu_1 \times (\mu_2 \times \mu_3)
  = (\mu_1 \times \mu_2) \times \mu_3 \,.
\]

\emph{Note:} A sketch of the proof is given in Lieb and Loss,
Corollary 1.11.  Write out a detailed argument.

\item (From Lieb and Loss.)  Give a ``counterexample'' to Tonelli's
  theorem in the absence of $\sigma$-finiteness.

\emph{Hint:} Take the Lebesgue measure on $[0,1]$ as one measure space
and the counting measure on $[0,1]$ as the other.

\item Give a ``counterexample'' to Fubini's
  theorem in the absence of summability.

\end{enumerate}

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