User:IntegralPython/sandbox/Fractal measure: Difference between revisions

Revision as of 15:17, 30 September 2020

Revision as of 11:30, 1 October 2020

Line 1:

'''Fractal measure''' is any [[measure (mathematics)|measure]] which generalizes the notions of length, area, and [[volume]] to non-[[integer]] dimensions, especially in application towards [[fractal]]s. There is no unique fractal measure, in part although not entirely due to the lack of a unique definition of [[fractal dimension]]; the most common fractal measures include the [[Hausdorff measure]] and the packing measure, based off of the [[Hausdorff dimension]] and [[packing dimension]] respectively.<ref>http://www.diva-portal.org/smash/get/diva2:22333/FULLTEXT01</ref> Fractal measures are measures in the sense of [[measure theory]], and are usually defined to agree with the ''n''-dimensional [[Lebesgue measure]] when ''n'' is an integer.<ref>https://link.springer.com/chapter/10.1007/978-1-4757-2958-0_1</ref> Fractal measure can be used to define the fractal dimension or vice versa. Although related, differing fractal measures are not equivalent, and may provide different measurements for the same shape.

A Carathéodory construction is a constructive method of building fractal measures, used to create [[measure (mathematics)| measure]]s from similarly defined [[outer measure]]s.

Although related, differing fractal measures are not the same, and may provide different measurements for the same shape.

==Carathéodory Construction==

Let ''τ''&nbsp;:&nbsp;Σ&nbsp;→&nbsp;[0,&nbsp;+∞] be a set function defined on a class Σ of subsets of ''X'' containing the empty set ∅, such that ''τ''(∅)&nbsp;=&nbsp;0. One can show that the set function ''μ'' defined by<ref>http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.517.5903&rep=rep1&type=pdf</ref>

:<math>\mu (E) = \lim_{\delta \to 0} \mu_{\delta} (E),</math>

where

:<math>H^d_\mu_{\delta} (SE) = \inf \left \{ \left. \sum_{i = 1}^{\infty} \tau (C_{i}) \right| C_{i} \in \Sigma, \operatornamemathrm{diam} U_i(C_{i})^d: \leq \delta, \bigcup_{i = 1}^{\infty} C_{i} U_i\supseteq S,E \operatorname{diam} U_i<\delta\right \},</math>

is not only an outer measure, but in fact a metric outer measure as well. (Some authors prefer to take a [[supremum]] over ''δ''&nbsp;&gt;&nbsp;0 rather than a [[Limit of a function|limit]] as ''δ''&nbsp;→&nbsp;0; the two give the same result, since ''μ''_''δ''(''E'') increases as ''δ'' decreases.)

The function and ___domain of ''τ'' may determine the specific measure obtained. For instance, if we give

:<math>\tau(C) = \mathrm{diam} (C)^s,\,</math>

where ''s'' is a positive constant and where ''τ'' is defined on the [[power set]] of all subsets of ''X'', the associated measure ''μ'' is the ''s''-dimensional [[Hausdorff measure]]. More generally, one could use any so-called [[dimension function]]. If instead ''τ'' is defined only on [[ball (mathematics)| ball]]s of ''X'', the associated measure is the [[spherical measure]].

This construction is how the Hausdorff and [[packing measure]]s are obtained.

==Hausdorff measure==

{{main|Hausdorff measure}}

The Hasudorff measure is the most-used fractal measure and provides a definition for [[Hausdorff dimension]], which is in turn one of the most frequently used definitions of fractal dimension. Intuitively, the Hausdorff measure canis be thought of asa covering the set by other sets, and taking the smallest possible measure of the coverings as the they approach zero.

Let <math>(X,\rho)</math> be a [[metric space]]. For any subset <math>U\subset X</math>, let <math>\mathrm{diam}\;U</math> denote its diameter, that is

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

Let <math>S</math> be any subset of <math>X,</math> and <math>\delta>0</math> a real number. We take

:<math>H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},</math>

where the infimum is over all countable covers of <math>S</math> by sets <math>U_i\subset X</math> satisfying <math> \operatorname{diam} U_i<\delta</math>; the Hausdorff measure <math>H^d(S)</math> is the limit of <math>H^d_\delta(S)</math> as <math>\delta</math> approaches zero.

When the ''d''-dimensional Hausdorff measure is an integer, <math>H^d(S)</math> is proportional to the [[Lebesgue measure]] for that dimension. Due to this, some definitions of Hausdorff measure include a scaling by the volume of the unit [[N-sphere|''d''-ball]], expressed using [[gamma function|Euler's gamma function]] as