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\begin{document}

\centerline {\bf Math 523a - Midterm - In class part}

\bigskip

\begin{enumerate}

\item Let $(X,\M,\mu)$ be a measure space. Let $f \in L^1(X,\M,\mu)$ 
\begin{enumerate}
\item 
Show that if there is a constant $\epsilon>0$ such that 
$f \ge \epsilon$ a.e., then $\mu(X)<\infty$. 
\item
Show that if $f>0$ a.e., then $\mu$ is $\sigma$-finite. 
(Recall that $\sigma$-finite means that there is
a countable collection of measurable sets $X_n$ such that 
$X=\cup_{n=1}^\infty X_n$ and $\mu(X_n)<\infty$.) 
\end{enumerate}

\item 
Let $(X,\M,\mu)$ be a measure space and $f \in L^1(X,\M,\mu)$. Find
\bea
\lim_{n \ra \infty} \int f(x) \, \exp(-n |f(x)-1|) \, d \mu
\eea
You should prove your result. 

\item 
Let $\mu$ be a finite Borel measure on $\reals^2$. Define a function on 
$\reals$ by 
\bea
G(t)=\mu(\{(x,y): x+y > t\})
\eea
Prove $G(t)$ is right continuous. 

\end{enumerate}

\end{document}
\end


\newpage

\item 
Let $\mu$ be a finite Borel measure on $\reals$. Define
$G(x)=\mu([x,\infty))$. Prove that $G$ is a decreasing, left-continuous
function on $\reals$. 

\item 
Let $(X,\M)$ be a measurable space. Let $f: X \ra \reals$. 
Prove that if $f^{-1}((-\infty,r)) \in \M$ for all rational numbers
$r$, then $f$ is measurable. 




\item 
Let $(X,\M,\mu)$ be a measure space with $\mu(X)<\infty$. Find 
\bea
\lim_{n \ra \infty} \int [\cos(f(x))]^{2n} \, d \mu
\eea

\item 
Let $(X,\M,\mu)$ be a measure space and $f \in L^1(X,\M,\mu)$. Find
\bea
\lim_{n \ra \infty} \int f(x) \, [\cos(f(x))]^{2n} \, d \mu
\eea


