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Let $\varepsilon>0,s\geq0$ and $C\subseteq \mathbb{R}^d$ be randomly given. Now define: $$ \mathcal{H}^s_\varepsilon(C)= \inf\biggl\{\,\sum^\infty_{n=1}(\rho (A_n))^s\biggm| C\subseteq \bigcup^\infty_{n=1}A_n, \rho(A_n)<\varepsilon\,\biggl\} $$ where $\rho(C)=\sup_{x,y\in C}|x-y|$.

Then the Hausdorff measure is given by $\mathcal{H}^s(C)=\lim_{\varepsilon \to 0} \mathcal{H}^s_\varepsilon(C)$.

Show that for any $C$ there exists a $s_0$ such that $$\mathcal{H}^s(C)=\begin{cases}+\infty, & s<s_0\\ 0, & s >s_0 \end{cases}$$ This $s_0$ is called the Hausdorff dimension of $C$. Can anyone help me with this exercise? I am completely stumped. If someone could at least give me a hint that would be wonderful.

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Try proving the following claims:

  • If $\mathcal H^s(C)>0$, then $\mathcal H^r(C)=\infty$ for all $r<s$.
  • If $\mathcal H^s(C)<\infty$, then $\mathcal H^r(C)=0$ for all $r>s$.

Assume now that the above claims have been proven. Show that $\mathcal H^{d+1}(\mathbb R^d)=0$ and thus $\mathcal H^{d+1}(C)=0$. (Sketch: Cover the cube $[0,1]^d$ with smaller cubes and use this cover to show that $\mathcal H^{d+1}([0,1]^d)=0$. Then use the fact that $\mathbb R^d$ is a union of countably many such cubes.) From the second claim above we get $\mathcal H^s(C)=0$ for all $s\geq d+1$. Therefore $s_0=\inf\{s\geq0;\mathcal H^s(C)=0\}$ is a number in $[0,\infty)$.

By the definition of $s_0$ we have $\mathcal H^s(C)=0$ for all $s>s_0$. Take then any $s<s_0$; we need to show that $\mathcal H^s(C)=\infty$. Take any number $r\in(s,s_0)$. It follows from the definition of $s_0$ that $\mathcal H^r(C)>0$. It then follows from the first claim above that $\mathcal H^s(C)=\infty$.

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  • $\begingroup$ I'm trying to use the fact that $\mathcal{H}^s(C)$ is a decreasing function in $s$ but I'm still running into some difficulty. Because if the sum converges a $s$ it's hard to say anything for an arbitrarily larger power. Could you maybe elaborate a bit more? $\endgroup$ – D. Vente Oct 3 '14 at 11:39
  • $\begingroup$ I added details about going from my two claims to the desired result. (Those two claims are useful properties of the Hausdorff measure.) Try to prove those claims. You can also find the claims and their proofs (or hints if they are exercises) in many books in measure theory. $\endgroup$ – Joonas Ilmavirta Oct 4 '14 at 17:02

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