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\begin{document}

\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 8, due Wednesday, November 5.
\end{large}
\end{center}

{\bf Name:} Insert your name here.

\begin{questions}

\question {\em (Combines Pugh's Ch. 6 \# 1-2)} Let $f:\R \to \R$ be $f(x)=ax+b$ for some $a,b \in \R$. Prove that
$m^\ast \circ f(A)=|a|\cdot m^\ast(A)$ for each $A \subset \R$, where $m^\ast$ is the Lebesgue outer measure on $\R$.
\begin{sol}
Insert answer here.
\end{sol}

\question Use the formula from the prior problem to show that the middle third Cantor set $C$
satisfies $m^\ast(C)=0$, where $m^\ast$ is Lebesgue outer measure. ({\em Hint:} Use the self-similarity.)
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Royden \S 2.2 \# 7) } A set of real numbers is said to be a $G_\delta$ set if it is the intersection of a countable collection of open sets. Show that for any bounded set $E$,
there is a $G_\delta$ set $G$ for which $E \subset G$ and $m^\ast(G)=m^\ast(E)$. 
\begin{sol}
Insert answer here.
\end{sol}

\question Fix some real number $d \geq 0$. For a subset $A \subset \R$ and $\delta>0$, let
$$H^d_\delta(A)= \inf \big\{ \sum_k |I_k|^d \big\},$$
where the infimum is taken over all countable covers $\{I_k\}$ of $A$ by open intervals each of which has length less than $\delta$. The {\em $d$-dimensional Hausdorff outer measure} of $A$ is
$$H^d(A)=\lim_{\delta \to 0} H^d_\delta(A).$$
You can use without proof that $H^d$ is an outer measure. You may also use without proof that when $d=1$, $H^d$ is the Lebesgue outer measure on $\R$.
\begin{parts}
\part Explain why if $\delta<\delta'$, then $H^d_\delta(A) \geq H^d_{\delta'}(A)$ for every $A \subset \R$. ({\em Remark}: It follows that $H^d(A)=\sup_{\delta>0} H^d_\delta(A)$.)
\begin{sol}
Insert answer here.
\end{sol}

\part Show that $H^d([0,1])=0$ for every $d > 1$.
\begin{sol}
Insert answer here.
\end{sol}

\part Let $L \in \R$ be positive, and let $k$ be a positive integer. Let 
$$\Delta=\{{\mathbf v} \in \R^k~:~\text{$v_i \geq 0$ and
$\sum_{i=1}^k v_i=L$}\}.$$
Fix a $d \in \R$ with $0 < d < 1$. Consider the function 
$$m:\Delta \to \R; \quad {\mathbf v} \mapsto \sum_{i=1}^k v_i^d.$$
Show that this function has a unique global maximum which is attained at the vector
where each $v_i=\frac{L}{k}.$
\begin{sol}
Insert answer here.
\end{sol}

\part Use the prior part to argue that $H^d([0,1])=\infty$ whenever $0<d<1$.
\begin{sol}
Insert answer here.
\end{sol}
\end{parts}

{\em Final remarks on this problem:} It can be shown that for any set $A \subset \R$, there
is a unique $0 \leq D \leq 1$ so that 
$$H^d(A)=\infty \quad \text{for $0 \leq d<D$} \qquad \text{and} \qquad H^d(A)=0 \quad \text{for $d>D$}.$$
This number $D$ is called the {\em Hausdorff dimension} of $A$. Once you do the exercises above, you will have shown that the Hausdorff dimension of $[0,1]$ is $1$.

If you would like a challenge, try to show that the Hausdorff dimension of the middle third Cantor set is $\frac{\log 2}{\log 3}$.

\end{questions}
\end{document}
