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\begin{document}

\centerline {\bf Math 523a - Homework 6} 

\bigskip

\begin{enumerate}

\item Folland, problem 4, p 88.

\item Folland, problem 12, p 92.

\item Folland, problem 16, p 92.

\item Folland, problem 17, p 93.

\item Folland, problem 21, p 94.

\item Consider the vector space $V$ of finite signed measures on 
some fixed measurable space $(X,\M)$. In class we defined a norm 
on it by \\ $||\mu||=|\mu|(X)$. 
\begin{enumerate}
\item Prove that $V$ is complete in this norm. 
\item Prove that 
\bea
||\mu|| = \sup \int f \, d \mu
\eea
where the $\sup$ is over all measurable real valued $f$  with $|f|\le 1$ a.e.
\end{enumerate}

\item Let $F(x)$ be a right continuous increasing function on $\reals$
and let $\mu$ be the associated Borel measure. (So $\mu((a,b])=F(b)-F(a)$.)
Suppose that $F$ has only a finite number of 
discontinuities (jumps) and in between the jumps it is $C^1$. 
($C^1$ means that on the open intervals between the jumps it is 
differentiable and the derivative is a continuous function.)
Let $m$ be Lebesgue measure on the real line. The Lebesgue-Radon-Nikodym
theorem says that there are measures $\lambda$ and $\rho$ such that
$\mu=\lambda+\rho$ with $\rho \perp m$ and $\lambda << m$. And there is 
a non-negative measurable function $f$ such that $\lambda=f m$. 
Give an explicit description of $f$ and $\rho$. You may assume the 
fundamental theorem of calculus: if $F$ is $C^1$ on $(a,b)$ and 
$a<c<d<b$, then $\int_c^d \, F^\prime(x) \, dx=F(d)-F(c)$. 

\end{enumerate}

\end{document}
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