Let the set $F \in {\Bbb{R}} ^n$. A $\delta$-cover of F is a countable or finite union of non-empty subsets $U_i \subset {\Bbb{R}}^n$ with diameter $\left\vert U_i \right\vert = \sup \{ \vert x-y\vert: x,y \in U_i \}$ less than $\delta$ which cover F:

F \subset \bigcup_{i=1}^{\infty} U_i.\end{displaymath}

Let $s \ge 0$ then

{\cal H}^s_{\delta} (F) =: \inf \left\{ \sum_{i=1}^{\infty}
 ...vert^s: \{U_i \} \mbox{is} \delta\mbox{--cover of }
F \right\}.\end{displaymath}

If $\delta$ gets smaller the number of allowed covers decreases and the infimum ${\cal H}^s_{\delta} (F)$ increases and has a limit for $\delta \rightarrow 0$:

{\cal H}^s (F) =: \lim_{\delta \rightarrow 0} {\cal H}^s_{\delta} (F),\end{displaymath}

This limit exists and is called the Hausdorff measure. If t>s and $\delta <1$ then

 \sum_i \vert U_i\vert^t \le \delta^{t-s} \sum_i \vert U_i\vert^s,

and ${\cal H}^t (F) \equiv 0$ if ${\cal H}^s (F) < \infty$.

So if you explore ${\cal H}^s (F)$ as a function of s there is a jump from $\infty$ to zero at some s which is called the Hausdorff dimension $\dim_H$ of F.

This document was generated using the LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997) Copyright © 1993, 1994, 1995, 1996, 1997, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

From the sci.fractals FAQ: A clear and concise (2 page) write-up of the definition of the Hausdorff-Besicovitch dimension in MS-Word 6.0 format is available in zip format. hausdorff.zip (~26KB) http://www.newciv.org/jhs/hausdorff.zip
Jürgen Dollinger