User:IntegralPython/sandbox/Fractal measure

Fractal measure is any measure which generalizes the notions of length, area, and volume to non-integer dimensions, especially in application towards fractals. 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. 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. Fractal measures find application in the study of fractal geometry, as well as in physics and biology through the study of fractal derivatives. Fractal measure can be used to define the fractal dimension or vice versa.

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

Hausdorff measure

Main article: 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 can be thought of as covering the set by other sets, and taking the smallest possible measure of the coverings as the they approach zero.

Let $(X,\rho )$ be a metric space. For any subset $U\subset X$, let $\mathrm {diam} \;U$ denote its diameter, that is

$\operatorname {diam} U:=\sup\{\rho (x,y):x,y\in U\},\quad \operatorname {diam} \emptyset :=0$

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

$H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},$

where the infimum is over all countable covers of $S$ by sets $U_{i}\subset X$ satisfying $\operatorname {diam} U_{i}<\delta$; the Hausdorff measure $H^{d}(S)$ is the limit of $H_{\delta }^{d}(S)$ as $\delta$ approaches zero.

When the d-dimensional Hausdorff measure is an integer, $H^{d}(S)$ 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 d-ball, expressed using Euler's gamma function as

${\frac {\pi ^{d/2}}{\Gamma ({\frac {d}{2}}+1)}}.$

Packing measure

Further information: 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. In contrast to Hausdorff measure, which covers the set being measured and can have intersecting covers, the packing measure requires no balls to intersect and becomes smaller than the Hausdorff measure.

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

$P_{0}^{s}(S)=\limsup _{\delta \downarrow 0}\left\{\left.\sum _{i\in I}\mathrm {diam} (B_{i})^{s}\right|{\begin{matrix}\{B_{i}\}_{i\in I}{\text{ is a countable collection}}\\{\text{of pairwise disjoint closed balls with}}\\{\text{diameters }}\leq \delta {\text{ and centres in }}S\end{matrix}}\right\}.$

The pre-measure is made into a true measure, where the s-dimensional packing measure of S is defined to be

$P^{s}(S)=\inf \left\{\left.\sum _{j\in J}P_{0}^{s}(S_{j})\right|S\subseteq \bigcup _{j\in J}S_{j},J{\text{ countable}}\right\},$

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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