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starting to include Catheodory construction: It's *super* messy and all over the pace right now, but I'm just trying to get the required information in before I start formatting it, along with still figuring out whether the construction should become the primary focus. It's definitely a work in progress |
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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'' : 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]]
<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==
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