en>ChrisGualtieri: /* Short lessons */Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-19T14:06:15Z

Short lessons: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

← Older revision		Revision as of 16:06, 19 December 2013
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	~~Friends~~ call ~~him Royal. She functions as~~ a ~~monetary officer~~ and ~~she will not change~~ it ~~whenever soon~~. The ~~preferred hobby~~ for ~~him~~ and ~~his kids~~ is ~~to perform badminton but he~~ is ~~having difficulties to find time~~ for it. ~~Delaware~~ is ~~our beginning location~~.<br><br>~~My webpage~~ :: ~~auto warranty (~~[http://~~Llollz~~.~~com~~/~~blogs~~/~~post~~/~~5238 official site~~])		In [[measure theory]], a field of mathematics, the '''Hausdorff density''' measures how concentrated a [[Radon measure]] is at some point.

			== Definition ==

			Let <math>\mu</math> be a Radon measure and <math>a\in\mathbb{R}^{n}</math> some point in [[Euclidean space]]. The ''s''-dimensional upper and lower Hausdorff densities are defined to be, respectively,
			:<math> \Theta^{*s}(\mu,a)=\limsup_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}</math>
			and
			:<math> \Theta_{*}^{s}(\mu,a)=\liminf_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}</math>
			where <math> B_{r}(a)</math> is the ball of radius ''r'' > 0 centered at ''a''. Clearly, <math>\Theta_{}^{s}(\mu,a)\leq \Theta^{s}(\mu,a)</math> for all <math>a\in\mathbb{R}^{n}</math>. In the event that the two are equal, we call their common value the '''s-density''' of <math>\mu</math> at ''a'' and denote it <math>\Theta^{s}(\mu,a)</math>.

			== Marstrand's theorem ==

			The following theorem states that the times when the ''s''-density exists are rather seldom.

			: '''Marstrand's theorem:''' Let <math>\mu</math> be a Radon measure on <math>\mathbb{R}^{d}</math>. Suppose that the ''s''-density <math>\Theta^{s}(\mu,a)</math> exists and is positive and finite for ''a'' in a set of positive <math>\mu</math> measure. Then ''s'' is an integer.

			== Preiss' theorem ==

			In 1987 Preiss proved a stronger version of Marstrand's theorem. One consequence is that that sets with positive and finite density are [[rectifiable set]]s.

			: '''Preiss' theorem:''' Let <math>\mu</math> be a Radon measure on <math>\mathbb{R}^{d}</math>. Suppose that ''m''<math>\geq 1</math> is an integer and the ''m''-density <math>\Theta^{m}(\mu,a)</math> exists and is positive and finite for <math>\mu</math> almost every ''a'' in the [[Support (measure theory)\|support]] of <math>\mu</math>. Then <math>\mu</math> is ''m''-rectifiable, i.e. <math>\mu\ll H^{m}</math> (<math>\mu</math> is [[absolutely continuous]] with respect to [[Hausdorff measure]] <math>H^m</math>) and the support of <math>\mu</math> is an ''m''-rectifiable set.

			==External links==
			* [http://www.encyclopediaofmath.org/index.php/Density_of_a_set Density of a set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
			* [http://www.encyclopediaofmath.org/index.php/Rectifiable_set Rectifiable set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

			== References ==
			* Pertti Mattila, ''Geometry of sets and measures in Euclidean spaces.'' Cambridge Press, 1995.

			* {{cite journal
			\| last = Preiss \| first = David
			\| title = Geometry of measures in <math>R^n</math>: distribution, rectifiability, and densities
			\| jstor = 1971410
			\| journal = Ann. Math. \| volume = 125
			\| issue = 3 \| pages = 537–643 \| year = 1987 \| doi=10.2307/1971410
			}}

			[[Category:Measure theory]]

en>Colin Sell at the piano: /* See also */ rm self-link

2012-06-30T20:26:05Z

Mean-shift - Revision history

en>ChrisGualtieri: /* Short lessons */Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

en>Colin Sell at the piano: /* See also */ rm self-link