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\begin{document}
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\vspace*{-.7in}

\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 9}

\noindent
\textbf{Posted: 4/6; Updated: 4/6; Due: Friday, 4/10/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: Chapter 11: Sections 8, 10-11. \smallskip \

\begin{prob}
  Prove that any compact set $K$ in $\nr^n$ is Lebesgue measurable and
  $m(K) < \infty$.
  \end{prob}

\begin{prob}
  Show that the Cantor set is measurable and has (Lebesgue) measure
  zero.
\end{prob}

\begin{prob}
  Let $c: [0,1] \to \nr$ be the Cantor function constructed in
  Homework 2. Extend $c$ to all $\nr$ by setting $c(x) := 0 $ for $x
  \le 0$ and $c(x) := 1$ for $x \ge 1$. Let $f(x) := c(x) + x $. Show
  that $f$ carries a set of (Lebesgue) measure zero (the Cantor set)
  to a set of positive measure. (This provides a counterexample, as
  one can show that $f$ is a homeomorphism of $\nr$, \emph{i.e}., a
  continuous map $\nr \to \nr$ which has a continuous inverse. You do
  not have to show that, though.)
\end{prob}

\begin{prob}
  A measure $\mu$ on a metric space with a given $\sigma$-algebra $\M$
  is called \emph{regular}, if the $\sigma$-algebra of Borel sets is a
  subalgebra of $\M$, and for every set $A$ in $\M$, we have
\[
\mu (A) = \inf \{ \mu (U) \; | \; U \text{ is open and } A \subset U \}
\]
and
\[
\mu (A) = \sup \{ \mu (K) \; | \; K \text{ is compact and } K \subset A \}.
\]
(1) Show that the counting measure on $\nz$ with the induced metric
from $\nr$ is regular. (2) Show that the delta measure with respect to
a point $x_0$ on any metric space is regular.
\end{prob}

\begin{prob}
  Show that the Lebesgue measure on $\nr^n$ is regular.
\end{prob}

\end{document}