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'''Fractal measure''' is a generalization of the concepts 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
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'' : Σ → [0, +∞] be a set function defined on a class Σ of subsets of ''X'' containing the empty set ∅, such that ''τ''(∅) = 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>\mu_{\delta} (E) = \inf \left\{ \left. \sum_{i = 1}^{\infty} \tau (C_{i}) \right| \begin{matrix} C_{i} \in \Sigma, \\ \mathrm{diam} (C_{i}) \leq \delta, \\ \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \end{matrix} \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'' > 0 rather than a [[Limit of a function|limit]] as ''δ'' → 0; the two give the same result, since ''μ''<sub>''δ''</sub>(''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.
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
:<math>\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}.</math><ref>https://encyclopediaofmath.org/wiki/Hausdorff_measure</ref>
==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'', ''d'') be a metric space with a subset ''S'' ⊆ ''X'' and let ''s'' ≥ 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==
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