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

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{{Userspace draft|date=March 2021}}

'''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.

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==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>

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:<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 thean [[spherical measure]].{{clarify|reason=The''s''-dimensional spherical measure is just the Borel measure on the sphere, and has nothing(not to do with what you are describing here. You are just saying that X can be coveredconfused by balls, and I think when you do this, you just getwith the ordinaryusual Hausdorff[[spherical measure]]), again. If covering by balls gives a different result than covering by other shapes, then how this works would have to be explained in detail. Anyway, just don't call it the sphericalfollowing measure cause it's notinequality that.}}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.}}

This construction is how the Hausdorff and [[packing measure]]s are obtained.{{clarify|reason=This statement is incorrect, the packing measure is NOT obtained in this way. In the above, you are covering the set to be measured, by contrast, the packing measure doe NOT cover, it jams balls into the inside, and only then uses the cartheodory extension theorem to get a measure. Don't confuse the covering used in the extension with the covering of the set itself...}}

==Hausdorff measure==