Hausdorff measure

Introduction

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

Given any fixed set $E\subset {ℝ}^n$ one can consider the measures ${\mathscr{H}}^{\alpha }(E)$ with $\alpha$ varying in $[0,+\mathrm{\infty })$. We will see that for a fixed set $E$ there exists at most one value $\alpha$ such that ${\mathscr{H}}^{\alpha }(E)$ is finite and positive; while for every other value $\beta$ one will have ${\mathscr{H}}^{\beta }(E)=0$ if $\beta >\alpha$ and ${\mathscr{H}}^{\beta }(E)=+\mathrm{\infty }$ if . For example, if $E$ is a regular $2$-dimensional surface then only ${\mathscr{H}}^2(E)$ (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 ${\mathscr{H}}^{\alpha }$ is naturally defined on every metric space $(X,d)$, not only on ${ℝ}^n$.

Definition

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

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

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

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

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

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

$${\mathscr{H}}_{\delta }^{\alpha }(E):=inf\{\sum _{j=0}^{\mathrm{\infty }}{\omega }_{\alpha }{\left(\frac{\mathrm{diam}(B_j)}{2}\right)}^{\alpha }:B_j\subset X,\bigcup _{j=0}^{\mathrm{\infty }}{B}_j\supset E,\mathrm{diam}(B_j)\le \delta \forall j=0,1,\mathrm{\dots }\}.$$ (1)

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

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

$${\mathscr{H}}^{\alpha }(E):=\underset{\delta \to 0^{+}}{lim}{\mathscr{H}}_{\delta }^{\alpha }(E).$$ (2)

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

Title	Hausdorff measure
Canonical name	HausdorffMeasure
Date of creation	2013-03-22 14:27:26
Last modified on	2013-03-22 14:27:26
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	8
Author	paolini (1187)
Entry type	Definition
Classification	msc 28A78
Related topic	HausdorffDimension
Related topic	LebesgueMeasure