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\begin{document}

\centerline {\bf Math 523b - Homework 2} 

\bigskip

\begin{enumerate}

\item Problem 56, p. 135 in Folland. 

\item Problem 60, p. 138 in Folland. 

\item Problem 63, p. 138 in Folland. 

\item Problem 67, p. 142 in Folland. 

\item Problem 68, p. 142 in Folland. 

\item Let $X$ be a locally compact Hausdorff space. Let $\mu$
be a Borel measure on $X$. We say it is a Radon measure if it satisfies
these three properties:
\begin{enumerate}
\item (local finiteness) For all compact sets $K$, $\mu(K) < \infty$. 
\item (outer regularity) For all Borel sets $E$
$$
\mu(E)=\inf \{ \mu(U) : U \, open, E \subset U \}
$$
\item (inner regularity) For all Borel sets $E$
$$
\mu(E)=\sup \{ \mu(K) : K \, compact , K \subset E \}
$$
\end{enumerate}
Prove that if $\mu$ is such a measure, then $C_c(X)$, the continuous
functions with compact support, are dense in $L^1(X,\mu)$.

\item Let $X$ be a separable compact metric space. Prove that 
$C(X)$ (real or complex valued) is separable. Hint: Let 
$x_k$ be a dense sequence in $X$ and consider the 
algebra of functions generated by the functions
$$
f_{k,n} = \max \{ 0, 1-n \rho(x,x_k)\}
$$
where $\rho$ is the metric. 

\end{enumerate}
\end{document}
\end

\item Let $f$ be a function on $[0,1]$ which is integrable with 
respect to Legesgue measure. Suppose that 
$$
\int_0^1 f(x) \, e^{nx} \, dx=0
$$
for $n=0,1,2,\cdots$.

\no (a) Prove that $f=0$ a.e. 

\no (b) Suppose we only have the above equation for $n=1,2,\cdots$. 
Does the conclusion still follow? Prove your answer. 



