\begin{assignment}{2022-09-07}{2022-09-14}
  \item
    \begin{parts}
      \item Say $\mu$ is a \emph{translation invariant} measure on $(\R^d, \mathcal L)$ (i.e. $\mu(x + A) = \mu(A)$ for all $A \in \mathcal L$, $x \in \R^d$) which is finite on bounded sets.
	Show that $\exists c \geq 0$ such that $\mu(A) = c \lambda(A)$.
      \item Let $T:\R^d \to \R^d$ be a linear transformation, and $A \in \mathcal L$.
	Show that $T(A) \in \mathcal L$ and $\lambda(T(A)) = \abs{\det(T)} \lambda(A)$.
	\par
	\notes{Hint: Express $T$ in terms of elementary transformations.}
    \end{parts}

  \item
    \begin{parts}
      \item
	Let $\mathcal E \subset \mathcal P(X)$, and $\rho: \mathcal E \to [0, \infty]$ be such that $\emptyset \in \mathcal E$, $\rho(\emptyset) = 0$.
	For any $A \subset X$ define
	$$
	  \mu^*(A) = \inf\Bigl\{ \sum_1^\infty \rho(E_i) \mathop\Big| E_i \in \mathcal E, \text{ and } A \subset \bigcup_{1}^\infty E_j \Bigr\}.
	$$
	Show that $\mu^*$ is an outer measure.

      \item
	Let $(X, d)$ be any metric space, $\delta > 0$, $\alpha \geq 0$ and define
	$$
	  \mathcal E_\delta = \{A \subset X \st \diam(A) < \delta \}
	  \quad\text{and}\quad
	  \rho_\alpha (A) = \frac{ \pi^{\alpha/2} }{ \Gamma( 1 + \frac{\alpha}{2} ) } \paren[\Big]{ \frac{\diam(A)}{2} }^\alpha.
	$$
	Let $H_{\alpha,\delta}^*$ be the outer measure obtained with $\rho = \rho_\alpha$ and the collection of sets $\mathcal E_\delta$.
	Define $H^*_\alpha = \lim_{\delta \to 0} H^*_{\alpha, \delta}$.
	Show $H^*_\alpha$ is an outer measure and restricts to a measure $H_\alpha$ on a $\sigma$-algebra that contains all Borel sets.
	The measure $H_\alpha$ is called the \emph{Hausdorff measure of dimension $\alpha$.}

      \item
	If $X = \R^d$, and $\alpha = d$ show that $H_d$ is a non-zero, finite constant multiple of the Lebesgue measure. \noteb{In fact $H_d = \lambda$ because of our choice of normalization constant, but the proof requires a little more work.}

      \item
	Let $S \in \mathcal B(X)$.
	Show that there exists (a unique) $d \in [0, \infty]$ such that $H_\alpha(S) = \infty$ for all $\alpha \in (0, d)$, and $H_\alpha(S) = 0$ for all $\alpha \in (d, \infty)$. 
	This number is called the \emph{Hausdorff dimension} of the set $S$.

      \item Compute the Hausdorff dimension of the Cantor set.
    \end{parts}

  \item
    Using notation from the previous question, let $\mathcal S_\delta = \{ B(x, r) \st x \in X, r \in (0, \delta) \}$.
    Using the collection of sets $\mathcal S_\delta$ and the function $\rho = \rho_\alpha$, we obtain an outer measure $S_{\alpha, \delta}^*$.
    As before one can show that $S_\alpha^* = \lim_{\delta \to 0} S_{\alpha, \delta}^*$ is an outer measure, and gives a Borel measure $S_\alpha$.
    \begin{parts}
      \item Do there exist $c_1, c_2 \in (0, \infty)$ such that $c_1 S_\alpha(A) \leq H_\alpha(A) \leq c_2 S_\alpha(A)$ for all Borel sets $A$?
	Prove or disprove it.
      \item If $X = \R^d$ with the standard metric show that $S_d \leq \lambda$.
	\\
	\noteb{%
	  You may assume $\rho_d( B_r ) = \lambda(B_r)$.
	  From the previous parts one can show $S_d = c \lambda$ for some $c > 0$.
	  We will return to this later and show $S_d = \lambda$.%
	}
      \itemopt \emph{(Optional Challenge)} Show by example $S_\alpha \neq H_\alpha$ in general.
    \end{parts}
\end{assignment}
