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| 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]].
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| ==Definition==
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| Let <math>(X,\rho)</math> be a metric space. For any subset <math>\scriptstyle U\subset X</math>, let <math>\mathrm{diam}\;U</math> denote its diameter, that is
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| :<math>\mathrm{diam}\;U :=\sup\{\rho(x,y)|x,y\in U\}, \quad \mathrm{diam}\;\emptyset:=0</math>
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| Let ''<math>S</math>'' be any subset of ''<math>X</math>'', and <math>\delta>0</math> a real number. Define
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| :<math>H^d_\delta(S)=\inf\Bigl\{\sum_{i=1}^\infty (\operatorname{diam}\;U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S,\,\operatorname{diam}\;U_i<\delta\Bigr\}.</math>
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| (The infimum is over all countable covers of ''<math>S</math>'' by sets <math>\scriptstyle U_i\subset X</math> satisfying <math>\scriptstyle \operatorname{diam}\;U_i<\delta</math>.)
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| Note that <math>\scriptstyle H^d_\delta(S)</math> is monotone decreasing in δ since the larger δ is, the more collections of sets are permitted, making the infimum smaller. Thus, the limit <math>\scriptstyle\lim_{\delta\to 0}H^d_\delta(S)</math> exists but may be infinite. Let
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| :<math> H^d(S):=\sup_{\delta>0} H^d_\delta(S)=\lim_{\delta\to 0}H^d_\delta(S).</math>
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| 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|Carathéodory-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.
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| 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>\scriptstyle H^d_\delta(S)</math> may be different {{harv|Federer|1969|loc=§2.10.2}}. If ''<math>X</math>'' 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.<ref>{{Citation|last = Yeh|first = J.|year = 2006|title = Real analysis: theory of measure and integration|page = 681}}</ref>
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| ==Properties of Hausdorff measures==
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| 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'',
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| :<math> \lambda_d(E) = 2^{-d} \alpha_d H^d(E)\,</math>
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| where α<sub>''d''</sub> is the volume of the unit [[N-sphere|''d''-ball]]; it can be expressed using [[gamma function|Euler's gamma function]]
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| :<math>\alpha_d =\frac{\Gamma(\frac12)^d}{\Gamma(\frac{d}{2}+1)} =\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}.</math>
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| '''Remark'''. Some authors adopt a definition of Hausdorff measure slightly different from 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.
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| ==Relation with Hausdorff dimension==
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| One of several possible equivalent definitions of the [[Hausdorff dimension]] is
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| :<math>
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| \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),
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| </math> | |
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| where we take
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| :<math>\inf\emptyset=\infty. \,</math>
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| ==Generalizations==
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| 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 <math>\scriptstyle\mathbb{R}^n</math> is said to be [[rectifiable set|''<math>m</math>''-rectifiable]] if it is the image of a [[bounded set]] in <math>\scriptstyle\mathbb{R}^n</math> under a [[Lipschitz function]]. If <math>m<n</math>, then the ''<math>m</math>''-dimensional Minkowski content of a closed ''<math>m</math>''-rectifiable subset of <math>\scriptstyle\mathbb{R}^n</math> is equal to <math>2^{-m}\alpha_m</math> times the ''<math>m</math>''-dimensional Hausdorff measure {{harv|Federer|1969|loc=Theorem 3.2.29}}.
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| 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:
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| :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 set function satisfying <math>\phi(\emptyset )=0</math>.
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| This is the Hausdorff measure of <math>S</math> with [[dimension function | 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>\scriptstyle H^\phi(S)\in(0,\infty)</math> with an appropriate <math>\phi.</math> Examples of gauge functions include <math>\scriptstyle \phi(t)=t^2\,\log\log\frac 1t</math> or <math>\scriptstyle\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>\scriptstyle\mathbb{R}^n</math> when <math>n>2</math>, and the latter when <math>n=2</math>.
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| == See also ==
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| <div style="-moz-column-count:4; column-count:4;"> | |
| * [[Hausdorff dimension]]
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| * [[Geometric measure theory]]
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| * [[Measure theory]]
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| * [[Outer measure]]
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| </div> | |
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Favard_measure Favard measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| ==References==
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| <references/>
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| * {{citation|first1=Lawrence C.|last1=Evans|first2=Ronald F.|last2=Gariepy|title=Measure Theory and Fine Properties of Functions|publisher=CRC Press|year=1992}}.
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| * {{citation|first=Herbert|last=Federer|authorlink=Herbert Federer|title=Geometric Measure Theory|publisher=Springer-Verlag|year=1969|isbn=3-540-60656-4}}.
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| * Yan Kun (2007), [http://adsabs.harvard.edu/abs/2007PrGeo..22..451Y Fractal Measure].
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| * {{citation|first=Felix|last=Hausdorff|authorlink=Felix Hausdorff|title= Dimension und äusseres Mass|journal=[[Mathematische Annalen]] |volume=79|year=1918|issue=1-2|pages=157–179|doi=10.1007/BF01457179}}.
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| * {{citation|first=Frank|last=Morgan|authorlink=Frank Morgan (mathematician)|title=Geometric Measure Theory|publisher=Academic Press|year=1988}}.
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| * {{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}}.
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| {{DEFAULTSORT:Hausdorff Measure}}
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| [[Category:Fractals]]
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| [[Category:Measures (measure theory)]]
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| [[Category:Metric geometry]]
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| [[Category:Dimension theory]]
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