en>Scm: /* Dual transform */ Resolved disambiguation needed

2014-10-07T20:40:09Z

Dual transform: Resolved disambiguation needed

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en>Slawekb: Undid revision 595578368 by 180.214.168.191 (talk)

2014-02-15T12:44:48Z

Undid revision 595578368 by 180.214.168.191 (talk)

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en>SoledadKabocha: remove disambiguation link "sinogram" because the correct meanng is this article itself

2014-01-14T23:57:26Z

remove disambiguation link "sinogram" because the correct meanng is this article itself

← Older revision		Revision as of 01:57, 15 January 2014
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	Hi generally. ~~My name~~ is ~~Consuelo Pelkey~~. ~~To play golf~~ is ~~among~~ the ~~many things I really most~~. ~~Georgia has always been my domicile~~. ~~She works as~~ a ~~debt collector~~ but ~~her promotion never comes~~. ~~Check out~~ the ~~latest news~~ on ~~his website~~: ~~https~~://~~www~~.~~youtube~~.~~com~~/~~watch?v~~=~~pns2W~~-~~mSPU8~~<br><br>~~My web site~~ - [~~https~~://www.~~youtube~~.~~com~~/~~watch?v~~=~~pns2W~~-~~mSPU8 Quantum Pendant Fake~~]		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 <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

			:<math>\mathrm{diam}\;U :=\sup\{\rho(x,y)\|x,y\in U\}, \quad \mathrm{diam}\;\emptyset:=0</math>

			Let ''<math>S</math>'' be any subset of ''<math>X</math>'', and <math>\delta>0</math> 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\supseteq S,\,\operatorname{diam}\;U_i<\delta\Bigr\}.</math>

			(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>.)

			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

			:<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\|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.

			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>

			==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]]; it can be expressed using [[gamma function\|Euler's gamma function]]
			:<math>\alpha_d =\frac{\Gamma(\frac12)^d}{\Gamma(\frac{d}{2}+1)} =\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}.</math>

			'''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.

			==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 <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}}.

			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 set function satisfying <math>\phi(\emptyset )=0</math>.

			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>.

			== See also ==
			<div style="-moz-column-count:4; column-count:4;">
			* [[Hausdorff dimension]]
			* [[Geometric measure theory]]
			* [[Measure theory]]
			* [[Outer measure]]
			</div>

			==External links==
			* [http://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
			* [http://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
			* [http://www.encyclopediaofmath.org/index.php/Favard_measure Favard measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

			==References==
			<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}}.
			* Yan Kun (2007), [http://adsabs.harvard.edu/abs/2007PrGeo..22..451Y Fractal Measure].
			* {{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}}.
			* {{citation\|first=Frank\|last=Morgan\|authorlink=Frank Morgan (mathematician)\|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}}.

			{{DEFAULTSORT:Hausdorff Measure}}
			[[Category:Fractals]]
			[[Category:Measures (measure theory)]]
			[[Category:Metric geometry]]
			[[Category:Dimension theory]]

en>FrescoBot: Bot: misplaced invisible LTR marks and minor changes

2012-07-22T23:53:24Z

Bot: misplaced invisible LTR marks and minor changes

New page

Hi generally. My name is Consuelo Pelkey. To play golf is among the many things I really most. Georgia has always been my domicile. She works as a debt collector but her promotion never comes. Check out the latest news on his website: https://www.youtube.com/watch?v=pns2W-mSPU8<br><br>My web site - [https://www.youtube.com/watch?v=pns2W-mSPU8 Quantum Pendant Fake]

Radon transform - Revision history

en>Scm: /* Dual transform */ Resolved disambiguation needed

en>Slawekb: Undid revision 595578368 by 180.214.168.191 (talk)

en>SoledadKabocha: remove disambiguation link "sinogram" because the correct meanng is this article itself

en>FrescoBot: Bot: misplaced invisible LTR marks and minor changes