# Hausdorff measure

## Introduction

Given a real number $\alpha\geq 0$ we are going to define a Borel external measure $\mathcal{H}^{\alpha}$ 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 $m$-dimensional regular surface then one will show that $\mathcal{H}^{m}(M)$ is the $m$-dimensional area of $M$. However, being an external measure, $\mathcal{H}^{m}$ 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 $m=n$, it turns out that the Hausdorff measure $\mathcal{H}^{n}$ 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 $\mathcal{H}^{\alpha}(E)$ with $\alpha$ varying in $[0,+\infty)$. We will see that for a fixed set $E$ there exists at most one value $\alpha$ such that $\mathcal{H}^{\alpha}(E)$ is finite and positive; while for every other value $\beta$ one will have $\mathcal{H}^{\beta}(E)=0$ if $\beta>\alpha$ and $\mathcal{H}^{\beta}(E)=+\infty$ if $\beta<\alpha$. For example, if $E$ is a regular $2$-dimensional surface then only $\mathcal{H}^{2}(E)$ (which is the area of the surface) may possibly be finite and different from $0$ while, for example, the volume of $E$ will be $0$ and the length of $E$ will be infinite.

This can be used to define the dimension of a set $E$ (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 $\mathcal{H}^{\alpha}$ is naturally defined on every metric space $(X,d)$, not only on $\mathbb{R}^{n}$.

## Definition

Let $(X,d)$ be a metric space. Given $E\subset X$ we define the diameter of $E$ as

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

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

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

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

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

 $\mathcal{H}^{\alpha}_{\delta}(E):=\inf\left\{\sum_{j=0}^{\infty}\omega_{\alpha% }\left(\frac{\mathrm{diam}(B_{j})}{2}\right)^{\alpha}\colon B_{j}\subset X,\ % \bigcup_{j=0}^{\infty}B_{j}\supset E,\ \mathrm{diam}(B_{j})\leq\delta\ \forall j% =0,1,\ldots\right\}.$ (1)

The infimum is taken over all possible enumerable families of sets $B_{0},B_{1},\ldots,B_{j},\ldots$ which are sufficiently small ($\mathrm{diam}B_{j}\leq\delta$) and which cover $E$.

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

 $\mathcal{H}^{\alpha}(E):=\lim_{\delta\to 0^{+}}\mathcal{H}^{\alpha}_{\delta}(E).$ (2)

The number $\mathcal{H}^{\alpha}(E)\in[0,+\infty]$ is called $\alpha$-dimensional Hausdorff measure of the set $E\subset X$.

Title Hausdorff measure HausdorffMeasure 2013-03-22 14:27:26 2013-03-22 14:27:26 paolini (1187) paolini (1187) 8 paolini (1187) Definition msc 28A78 HausdorffDimension LebesgueMeasure