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Hausdorff measure

(Definition)

Introduction

Given a real number $\alpha\ge 0$ we are going to define a Borel external measure $\H ^\alpha$ on $\mathbb{R}^n$ with values in $[0,+\infty]$ which will comprehend and generalize the concepts of length (for $\alpha=1$ ), area ( $\alpha=2$ ) and volume ( $\alpha=3$ ) of sets in $\mathbb{R}^n$ . In particular if $M\subset \mathbb{R}^n$ is an -dimensional regular surface then one will show that $\H ^m(M)$ is the -dimensional area of . However, being an external measure, $\H ^m$ is defined not only on regular surfaces but on every subset of $\mathbb{R}^n$ thus generalizing the concepts of length, area and volume. In particular, for , it turns out that the Hausdorff measure $\H ^n$ is nothing else than the Lebesgue measure of $\mathbb{R}^n$ .

Given any fixed set $E\subset \mathbb{R}^n$ one can consider the measures $\H ^\alpha(E)$ with $\alpha$ varying in $[0,+\infty)$ . We will see that for a fixed set there exists at most one value $\alpha$ such that $\H ^\alpha(E)$ is finite and positive; while for every other value $\beta$ one will have $\H ^\beta(E)=0$ if $\beta>\alpha$ and $\H ^\beta(E)=+\infty$ if $\beta<\alpha$ . For example, if is a regular -dimensional surface then only $\H ^2(E)$ (which is the area of the surface) may possibly be finite and different from 0 while, for example, the volume of will be 0 and the length of will be infinite.

This can be used to define the dimension of a set (this is called the Hausdorff dimension). A very interesting fact is the existence of sets with dimension $\alpha$ which is not integer, as happens for most fractals.

Also, the measure $\H ^\alpha$ is naturally defined on every metric space , not only on $\mathbb{R}^n$ .

Definition

Let

be a metric space. Given $E\subset X$ we define the diameter of

$\displaystyle \mathrm{diam}(E) := \sup_{x,y\in E} d(x,y).$

Given a real number $\alpha$ we consider the conventional constant

$\displaystyle \omega_\alpha=\frac{\pi^{\alpha/2}}{\Gamma(\alpha/2+1)}$

where $\Gamma(x)$ is the gamma function.

For all $\delta>0$ , $\alpha\ge 0$ and $E\subset X$ let us define

$\displaystyle \H ^\alpha_\delta(E) := \inf \left\{\sum_{j=0}^\infty \omega_\alp... ...ty B_j\supset E,\ \mathrm{diam}(B_j) \le \delta\ \forall j=0,1,\ldots \right\}.$

(1)

The infimum is taken over all possible enumerable families of sets $B_0, B_1, \ldots, B_j,\ldots$ which are sufficiently small ( $\mathrm{diam}B_j \le \delta$ ) and which cover

Notice that the function $\H ^\alpha_\delta(E)$ is decreasing in $\delta$ . In fact given $\delta'>\delta$ the family of sequences considered in the definition of $\H ^\alpha_{\delta'}$ contains the family of sequences considered in the definition of $\H ^\alpha_\delta$ and hence the infimum is smaller. So the limit in the following definition exists: