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\def\course{MAT 324}
\def\semester{Fall 2007}

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\begin{center}
{\bf  PROBLEM SET 4}
\end{center}

\begin{enumerate}

\item Compute the integral of the Cantor-Lebesgue function $\int_0^1 F(x) dm$ from 
      Chapter 2.
 
\item If $f_n(x) =\sin^2(nx)$ find both $\int_0^{2\pi} \liminf_{n \to \infty} f_n(x) dm$ 
     and $\liminf_{n \to \infty}  \int_0^{2 \pi} f_n(x) dm$.

\item What is $\lim_{n \to \infty} \int_{-\infty}^\infty x^n e^{-n|x|} dm$? Find 
      the limit and prove it is correct.


\item  Suppose $\{f_n\}$ is a sequence of functions that converges 
      almost everywhere to a function $f$  and define  $F_n=\sup_{k=1,\dots,n} |f_n|$.
      Show that if  the integrals of $F_n$ remain bounded as $n \to \infty$ 
      then $\lim_n \int f_n dm = \int f dm$.

\item  Given a measureable function $f$ define its ``maximal function'' 
       as 
$$Mf(x) = \sup_{t >0} \frac 1{2t} \int_{x-t}^{x+t} |f(y)| dm(y).$$
       If $f$ is integrable on the reals, does $Mf$ have to be integrable?

\item Show that $\sum_{n=1}^\infty \cos^n(2^n x)$ converges for  a.e. 
      $x$, but diverges on a dense set of $x$'s.


\end{enumerate}

\end{document}
