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\begin{document}

\centerline {\bf Math 523a - Midterm - Take home part}

\bigskip

\noindent {\bf Due Monday, Oct 22 at 10 am.} 

\noindent {\bf Rules:}
You may not talk to anyone about the exam. If the problem is 
unclear you can ask me for clarification, but I will not give hints.
You may refer to your class notes and Folland. But you cannot
use any other books or the web.

\noindent {\bf Exposition} is important. I will take off points if 
your solution is not clearly explained or contains irrelevant arguments. 

\bigskip




\begin{enumerate}

\item Recall that for two sets $E,F$, we define 
$E \Delta F = (E \setminus F) \cup (F \setminus E)$. 
And for a subset $E$ of $\reals$ we define 
$E+x = \{y+x: y \in E\}$. Let $m$ be Lebesgue measure on the real line.
Let $E \subset \reals$ be a Lebesgue measurable set with $m(E)<\infty$. 
Find the limit 
\bea
\lim_{x \ra \infty} m(E \Delta (E+x))
\eea
You should prove your answer.

\item 
Let $(X,\M)$ be a measurable space, and $(Y,d)$ a {\bf separable} metric space. 
Equip $Y$ with the Borel $\sigma$-algebra. Let $f_n : X \ra Y$ 
be measurable. Let $E \subset X$ be the set of $x$ such that 
$f_n(x)$ is a Cauchy sequence in $(Y,d)$. Prove that $E$ is 
measurable. (You may not assume that $Y$ is a complete metric space.)

\item 
Let $(X,\M,\mu)$ be a measure space. Let $f$ be a non-negative function in 
$L^1(X,\M,\mu)$ such that $\mu(\{x: f(x) \le 1\}) < \infty$. 

\noindent (a) Show that for positive integers $n$, $f^{1/n}$ is in $L^1$. 

\noindent (b) Find 
\bea
\lim_{n \ra \infty} \int \, f^{1/n} \, d\mu 
\eea
You should prove your answer. 

\newpage

\item 
Let $(X,\M,\mu)$ be a measure space with $\mu(X) < \infty$. 
For real-valued measurable functions $f,g$ on $X$, define
\bea
\rho(f,g) = \int \frac{|f-g|}{1 + |f-g|} \, d \mu 
\eea
Prove that $f_n \ra f$ in measure if and only if $\rho(f_n,f) \ra 0$. 

\item Let $C[0,1]$ be the set of real-valued continuous functions
on $[0,1]$. Define a metric on this set by 
\bea
\rho(f,g)=\int_{[0,1]} \, |f-g| \, dm
\eea
where $m$ is Lebesgue measure on $[0,1]$. 
Let 
\bea 
A=\{f \in C[0,1]: |f(x)| \le 1 \, \forall x\}
\eea
Find the interior of $A$. You should prove your answer. 

\end{enumerate}

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