Hausdorff measure
504109
223628952
2008-07-04T23:43:04Z
Sullivan.t.j
2036293
refined category
In [[mathematics]] a '''Hausdorff measure''' is a type of [[outer measure]], named for [[Felix Hausdorff]], that assigns a number in [0,∞] to each set in '''R'''<sup>''n''</sup> or, more generally, in any [[metric space]]. The zero dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. The one dimensional Hausdorff measure of a [[simple curve]] in '''R'''<sup>''n''</sup> is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a [[Lebesgue measure#Construction of the Lebesgue measure|measurable subset]] of '''R'''<sup>2</sup> is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0 which is not necessarily an integer. These measures are fundamental in [[geometric measure theory]]. They appear naturally in [[harmonic analysis]] or [[potential theory]].
==Definition==
Let (''X'',ρ) be a metric space. For any subset ''U'' ⊂ ''X'', let diam(''U'') denote its diameter, that is
that is diam(''U'') = sup{ ρ(''x'',''y'') | ''x'',''y'' ∈ ''U''}.
Let ''S'' be any subset of ''X'', and δ > 0 a real number. Define
:<math>H^d_\delta(S)=\inf\Bigl\{\sum_{i=1}^\infty \operatorname{diam}(U_i)^d: \bigcup_{i=1}^\infty U_i\supset S,\,\operatorname{diam}(U_i)<\delta\Bigr\}.</math>
(The infimum is over all countable covers of ''S'' by sets <math>U_i\subset X</math> satisfying <math>\operatorname{diam}(U_i)<\delta</math>.)
Note that <math>H^d_\delta(S)</math> is monotone decreasing in δ since the larger δ is, the more collections of sets are permitted. Thus, the limit <math>\lim_{\delta\to 0}H^d_\delta(S)</math> exists. Let
:<math> H^d(S):=\sup_{\delta>0} H^d_\delta(S)=\lim_{\delta\to 0}H^d_\delta(S).</math>
It can be seen that <math> H^d(S)</math> is an [[outer measure]] (more precisely, it is a [[metric outer measure]]). By general theory, its restriction to the σ-field of [[Outer_measure#Formal_definitions|Caratheodory-measurable sets]] is a measure. It is called the <math>d</math>-'''dimensional Hausdorff measure''' of <math>S</math>. Due to the [[metric outer measure]] property, all [[Borel subset|Borel]] subsets of <math>X</math> are <math>H^d</math> measurable.
In the above definition the sets in the covering are arbitrary. However, they may be taken to be open or closed, and will yield the same measure, although the approximations <math>H^d_\delta(S)</math> may be different {{harv|Federer|1969|loc=§2.10.2}}. If ''X'' is a [[normed space]] the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different measure.
==Properties of Hausdorff measures==
Note that if ''d'' is a positive integer, the ''d'' dimensional Hausdorff measure of '''R'''<sup>d</sup> is a rescaling of usual ''d''-dimensional [[Lebesgue measure]] <math>\lambda_d</math> which is normalized so that the Lebesgue measure of the unit cube [0,1]<sup>''d''</sup> is 1. In fact, for any Borel set ''E'',
:<math> \lambda_d(E) = 2^{-d} \alpha_d H^d(E)\,</math>
where α<sub>''d''</sub> is the volume of the unit [[N-sphere|''d''-ball]],
:<math>\alpha_d = \frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}.</math>
'''Remark'''. Some authors adopt a slightly different definition of Hausdorff measure than the one chosen here, the difference being that it is normalized in such a way that Hausdorff ''d''-dimensional measure in the case of Euclidean space coincides exactly with Lebesgue measure.
==Relation with Hausdorff dimension==
One of several possible equivalent definitions of the [[Hausdorff dimension]] is
:<math>\operatorname{dim}_{\mathrm{Haus}}(S):= \inf\{d\ge 0:H^d(S)=0\}=\sup\bigl(\{d\ge 0:H^d(S)=\infty\}\cup\{0\}\bigr),</math>
where we take <math>\inf\emptyset=\infty</math>.
==Generalizations==
In [[geometric measure theory]] and related fields, the [[Minkowski content]] is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of '''R'''<sup>''n''</sup> is said to be [[rectifiable set|''m''-rectifiable]] if it is the image of a [[bounded set]] in '''R'''<sup>''m''</sup> under a [[Lipschitz function]]. If ''m'' < ''n'', then the ''m''-dimensional Minkowski content of a closed ''m''-[[rectifiable]] subset of '''R'''<sup>''n''</sup> is equal to 2<sup>−m</sup>α<sub>''m''</sub> times the ''m''-dimensional Hausdorff measure {{harv|Federer|1969|loc=Theorem 3.2.29}}.
In [[fractal geometry]], some fractals with Hausdorff dimension <math>d</math> have zero or infinite <math>d</math>-dimensional Hausdorff measure. For example, [[almost surely]] the image of planar [[Brownian motion]] has Hausdorff dimension 2 and its two-dimensional Hausdoff measure is zero. In order to “measure” the “size” of such sets, mathematicians have considered the following variation on the notion of the Hausdorff measure.
In the definition of the measure <math>|U_i|^d</math> is replaced with <math>\phi(|U_i|)</math> where <math>\phi</math> is any monotone increasing function <math>\phi:[0,\infty)\to[0,\infty)</math> satisfying <math>\phi(0)=0</math>.
This is the '''Hausdorff measure''' of <math>S</math> with '''gauge function''' <math>\phi</math> or <math>\phi</math>-Hausdorff measure. A <math>d</math>-dimensional set <math>S</math> may satisfy <math>H^d(S)=0</math>, but <math>H^\phi(S)\in(0,\infty)</math> with an appropriate <math>\phi.</math> Examples of gauge functions include <math>\phi(t)=t^2\,\log\log\frac 1t</math> or <math>\phi(t) = t^2\log\frac{1}{t}\log\log\log\frac{1}{t}</math>. The former gives almost surely positive and <math>\sigma</math>-finite measure to the Brownian path in <math>\mathbb{R}^n</math> when <math>n>2</math>, and the latter when <math>n=2</math>.
==References==
* {{citation|first1=Lawrence C.|last1=Evans|first2=Ronald F.|last2=Gariepy|title=Measure Theory and Fine Properties of Functions|publisher=CRC Press|year=1992}}.
* {{citation|first=Herbert|last=Federer|authorlink=Herbert Federer|title=Geometric Measure Theory|publisher=Springer-Verlag|year=1969|isbn=3-540-60656-4}}.
* {{citation|first=Felix|last=Hausdorff|authorlink=Felix Hausdorff|title=Dimension und äusseres Mass|journal=Mathematische Annalen|volume=79|year=1918|pages=157-179}}.
* {{citation|first=Frank|last=Morgan|title=Geometric Measure Theory|publisher=Academic Press|year=1988}}.
* {{citation|first=E|last=Szpilrajn|authorlink=Edward Marczewski|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28111.pdf|title=La dimension et la mesure|journal=Fundamenta Mathematicae|volume=28|year=1937|pages=81-89}}.
[[Category:Fractals]]
[[Category:Measures (measure theory)]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]