Hausdorff measure


Given a real number α0 we are going to define a Borel external measureMathworldPlanetmath α on n with values in [0,+] which will comprehend and generalize the concepts of length (for α=1), area (α=2) and volume (α=3) of sets in n. In particular if Mn is an m-dimensional regular surface then one will show that m(M) is the m-dimensional area of M. However, being an external measure, m is defined not only on regular surfaces but on every subset of n thus generalizing the concepts of length, area and volume. In particular, for m=n, it turns out that the Hausdorff measureMathworldPlanetmath n is nothing else than the Lebesgue measureMathworldPlanetmath of n.

Given any fixed set En one can consider the measures α(E) with α varying in [0,+). We will see that for a fixed set E there exists at most one value α such that α(E) is finite and positive; while for every other value β one will have β(E)=0 if β>α and β(E)=+ if β<α. For example, if E is a regular 2-dimensional surface then only 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 dimensionMathworldPlanetmath). A very interesting fact is the existence of sets with dimension α which is not integer, as happens for most fractalsMathworldPlanetmath.

Also, the measure α is naturally defined on every metric space (X,d), not only on n.


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


Given a real number α we consider the conventional constant


where Γ(x) is the gamma functionDlmfDlmfMathworldPlanetmath.

For all δ>0, α0 and EX let us define

δα(E):=inf{j=0ωα(diam(Bj)2)α:BjX,j=0BjE,diam(Bj)δj=0,1,}. (1)

The infimum is taken over all possible enumerable families of sets B0,B1,,Bj, which are sufficiently small (diamBjδ) and which cover E.

Notice that the functionMathworldPlanetmath δα(E) is decreasing in δ. In fact given δ>δ the family of sequences Bj considered in the definition of δα contains the family of sequences considered in the definition of δα and hence the infimum is smaller. So the limit in the following definition exists:

α(E):=limδ0+δα(E). (2)

The number α(E)[0,+] is called α-dimensional Hausdorff measure of the set EX.

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