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'''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 measures find application in the study of fractal geometry, as well as in [[physics]] and [[biology]] through the study of [[fractal derivative]]s.<ref>{{cite journal |first=W. |last=Chen |title=Time–space fabric underlying anomalous diffusion |journal=Chaos, Solitons and Fractals |volume=28 |issue= 4|pages=923–929 |year=2006 |doi= 10.1016/j.chaos.2005.08.199|url=|arxiv=math-ph/0505023 |bibcode=2006CSF....28..923C |s2cid=18369880 }}</ref> Fractal measure can be used to define the fractal dimension or vice versa.

'''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 [[hausdorffHausdorff measure]] and the packing measure, based off of the [[hausdorffHausdorff 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 measuresmeasure findcan applicationbe inused theto studydefine ofthe fractal geometry,dimension asor wellvice asversa. inAlthough [[physics]]related, anddiffering [[biology]] through the study of [[fractal derivative]]s.<ref>{{citemeasures journalare |first=W.not |last=Chen |title=Time–space fabric underlying anomalous diffusion |journal=Chaosequivalent, Solitons and Fractalsmay |volume=28provide |issue=different 4|pages=923–929measurements |year=2006 |doi= 10.1016/j.chaos.2005.08.199|url=|arxiv=math-ph/0505023 |bibcode=2006CSF....28..923C |s2cid=18369880 }}</ref> Fractal measure can be used to definefor the fractalsame dimension or vice versashape.

Although related, differing fractal measures are not the same, 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.

==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 name="Geometric Outer-Measures">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| \begin{matrix} C_{i} \in \Sigma, \\ \operatornamemathrm{diam} U_i(C_{i})^d: \leq \delta, \\ \bigcup_{i = 1}^{\infty} C_{i} U_i\supseteq S,E \operatornameend{diammatrix} 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'' (i.e., <math>\Sigma = 2^X</math>), 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 <math>S^s</math> is an ''s''-dimensional spherical measure (not to be confused with the usual [[spherical measure]]), the following inequality applies:

<math>H^s(E) \le S^s(E) \le 2H^s(E)</math>.<ref name="Geometric Outer-Measures"/> {{clarify|reason=Covering by balls gives a different result than covering by general shapes; how this works needs to be explained in detail.}}

==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 is a 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. The outer-Hausdorff measure of <math>S</math> is

:<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>, and 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

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==Packing measure==

{{further|Packing dimension}}

Just as the packing dimension is in some ways a dual to the Hausdorff dimension, the packing measure is a counterpart to the Hausdorff measure. The packing measure is defined informally as the measure of "packing" a set with open balls, and calculating the measure of those balls.

Let (''X'',&nbsp;''d'') be a metric space with a subset ''S''&nbsp;⊆&nbsp;''X'' and let ''s''&nbsp;≥&nbsp;0. We take a "pre-measure" of ''S'', defined to be

:<math>P_0^s (S) = \limsup_{\delta \downarrow 0}\left\{ \left. \sum_{i \in I} \mathrm{diam} (B_i)^s \right| \begin{matrix} \{ B_i \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint closed balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}.</math><ref>https://projecteuclid.org/download/pdf_1/euclid.rae/1214571371</ref>

The [[pre-measure]] is made into a true [[measure (mathematics)|measure]], where the ''s'''''-dimensional packing measure''' of ''S'' is defined to be

:<math>P^s (S) = \inf \left\{ \left. \sum_{j \in J} P_0^s (S_j) \right| S \subseteq \bigcup_{j \in J} S_j, J \text{ countable} \right\},</math>

i.e., the packing measure of ''S'' is the [[infimum]] of the packing pre-measures of countable covers of ''S''.

==References==