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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 12} \vs \vs
\end{center}

\noindent
Throughout $\fM$ is a $\sigma$-algebra in a set $X$. As usual, the acronym AC stands for ``absolutely continuous.'' \vs \vs 

\noindent
{\bf Problem 1.} Suppose $\mu,\lambda$ are complex measures on $\fM$, and $E \in \fM$. Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
$|\mu+\lambda|(E) \leq |\mu|(E) + |\lambda|(E)$. \vs

\item[(ii)]
$\ds{|\mu|(E) = \sup \Big\{ \sum_{i=1}^n |\mu(E_i)| : E_1, \ldots, E_n \ \text{is a finite partition of} \ E \Big\} }$. \vs 
\end{enumerate}

\noindent
{\bf Problem 2.} Consider the relation $\ll$ on the space of finite positive measures on $\fM$. \vs
 
\begin{enumerate}
\item[(i)]
Prove that $\ll$ is transitive, and if $\mu \ll \lambda$ and $\lambda \ll \nu$, then the chain rule 
$$
\frac{d\mu}{d\nu} = \frac{d\mu}{d\lambda} \ \frac{d\lambda}{d\nu}   
$$
holds $\nu$-a.e. in $X$. \vs 

\item[(ii)]
If $\mu \ll \lambda$ and $\lambda \ll \mu$ (such pairs of measures are said to be {\bit equivalent}), how do the Radon-Nikodym derivatives $d\mu/d\lambda$ and $d\lambda/d\mu$ relate? \vs 
\end{enumerate}

\noindent
{\bf Problem 3.} Let $m$ and $\mu$ denote Lebesgue measure and the counting measure on $\RR$, respectively. \vs
\begin{enumerate}
\item[(i)]
Show that despite $m \ll \mu$ there is no $f$ for which $dm= f \, d\mu$. \vs

\item[(ii)]
Show that $\mu$ has no Lebesgue decomposition with respect to $m$. \vs 
\end{enumerate}
Why don't these failures contradict the Lebesgue-Radon-Nikodym theorem? \vs \vs  

\noindent
{\bf Problem 4.} Prove that the Jordan decomposition of a signed measure is minimal: If $\nu$ is a signed measure on $\fM$ and if $\nu = \mu - \lambda$ for some finite positive measures $\mu, \lambda$ on $\fM$, then $\mu \geq \nu^+$ and $\lambda \geq \nu^-$. (Hint: Use the Hahn decomposition theorem.) \vs \vs    

\noindent
{\bf Problem 5.} Suppose $f:[a,b] \to \RR$ is AC and $f' \in L^p[a,b]$ for some $1<p<\infty$. Show that $f$ is H\"{o}lder continuous of exponent $1/q$, where $1/p+1/q=1$. \vs \vs

\noindent
{\bf Problem 6.} Construct a homeomorphism $f:[0,1] \to [0,1]$ such that $f'(x)=0$ for almost every $x \in [0,1]$. (Hint: You may want to think about an infinite sum of suitably scaled Cantor functions.) \vs \vs 

\noindent
{\bf Problem 7.} Show that if $f,g$ are AC on $[a,b]$, so is their product $fg$. Use this to prove the integration by parts formula
$$
\int_a^b f(x) g'(x) \, dx = f(b)g(b)-f(a)g(a) - \int_a^b f'(x) g(x) \, dx. \vs
$$ 

\noindent
{\bf Problem 8.} Recall that a function $f : [0,1] \to \RR$ is {\bit $M$-Lipschitz} if $|f(x)-f(y)| \leq M |x-y|$
for all $x,y \in [0,1]$. Prove that $f$ is $M$-Lipschitz if and only if there exists a sequence $\{ f_n \}$ of continuously differentiable functions defined on $[0,1]$ such that \vs
\begin{enumerate}
\item[(i)]
$|f'_n(x)| \leq  M$ for all $n$ and all $x \in [0,1]$, and \vs
\item[(ii)]
$f_n(x) \to  f(x)$ for all $x \in [0,1]$ as $n \to \infty$.
\end{enumerate}
\end{document}

