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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 4} \vs \\
{\bf due Friday 3/7/2014} \vs \vs
\end{center}

\noindent
Unless otherwise stated, $X$ is a metric space. \vs \vs

\noindent
{\bf Problem 1.} A function $f: X \to [-\infty,+\infty]$ is called {\bit lower semi-continuous (l.s.c.)} if $f^{-1}(a,+\infty]$ is open for every $a \in \RR$, and {\bit upper semi-continuous (u.s.c.)} if $f^{-1}[-\infty,a)$ is open for every $a \in \RR$. Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
$f$ is continuous iff it is both l.s.c. and u.s.c. \vs

\item[(ii)]
The characteristic function $\chi_E$ is l.s.c. iff $E$ is open, and is u.s.c. iff $E$ is closed. \vs

\item[(iii)]
Local formulation of semi-continuity:
\begin{align*}
f \ \text{is l.s.c.} & \Longleftrightarrow \liminf_{x \to p} f(x) \geq f(p) \ \text{for every} \ p \in X \\
f \ \text{is u.s.c.} & \Longleftrightarrow \limsup_{x \to p} f(x) \leq f(p) \ \text{for every} \ p \in X. \vs
\end{align*}

\item[(iv)]
If $f_\alpha$ is l.s.c. (resp. u.s.c.) for each $\alpha$, so is $\sup_\alpha f_\alpha$ (resp. $\inf_\alpha f_\alpha$). Here $\alpha$ runs over an arbitrary index set. \vs

\item[(v)]
If $X$ is compact, a l.s.c. (resp. u.s.c.) function $X \to \RR$ reaches its minimum (resp. maximum) at some point of $X$. \vs

\item[(vi)]
If $X$ is locally compact and $f:X \to [0,+\infty]$ is l.s.c., then  
$$
f(x)= \sup \ \{ g(x): g \in \sC_c(X) \ \text{and} \ 0 \leq g \leq f \}. 
$$
\end{enumerate}
(Hint for (vi): Use Urysohn's lemma.) \vs

\noindent
{\bf Problem 2.} Recall that the space $\sC[0,1]$ of continuous complex-valued 
functions on $[0,1]$, equipped with the distance
$$
\| f-g \|_{\infty} = \sup_{x \in [0,1]} |f(x)-g(x)|,
$$  
is a complete separable metric space. \vs
\begin{enumerate}
\item[(i)]
Show that the closed unit ball $\{ f \in \sC[0,1]: \| f \|_{\infty} \leq 1 \}$ is not compact. \vs

\item[(ii)]
Show that every compact subset of $\sC[0,1]$ has empty interior. Conclude that
$\sC[0,1]$ is not locally compact. \vs 

\item[(iii)]
Show that $\sC[0,1]$ is not $\sigma$-compact. \vs
\end{enumerate}
(Hint: For (i), find a sequence in the closed unit ball with no convergent subsequence. For (iii), use Baire's theorem.) \vs 

\newpage
\noindent
{\bf Problem 3.} Describe the compact sets $K \subset \RR$ for which there is an $f \in \sC_c(\RR)$ with $\supp(f)=K$. \vs 

\noindent
{\bf Problem 4.} Show that the sum of two Radon measures is Radon. \vs 

\noindent
{\bf Problem 5.} Equip $\RR$ with the discrete metric (so all subsets are now open) and let $\{ r_n \}_{n \geq 1}$ be an enumeration of $\QQ$. Is the measure $\mu:2^{\RR} \to [0,+\infty)$ defined by  
$$
\mu(E) = \sum_{n: r_n \in E} 2^{-n}
$$  
a Radon measure? \vs

\noindent
{\bf Problem 6.} {\it (Intermediate value property of measures)} Suppose $\mu$ is a Radon measure on $X$ with the property that $\mu(\{ x \})=0$ for every $x \in X$. Let $E \subset X$ be a Borel set with $0<\mu(E)<+\infty$. Prove that for every $0<a<\mu(E)$ there is a Borel set $A \subset E$ such that $\mu(A)=a$. (Hint: Every point of $X$ has neighborhoods of arbitrarily small measure.)



\end{document}
