User:IntegralPython/sandbox/Fractal measure

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 byhttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.517.5903&rep=rep1&type=pdf

:$\backslash mu\; (E)\; =\; \backslash lim\_\{\backslash delta\; \backslash to\; 0\}\; \backslash mu\_\{\backslash delta\}\; (E),$

where

:$\backslash mu\_\{\backslash delta\}\; (E)\; =\; \backslash inf\; \backslash left\backslash \{\; \backslash left.\; \backslash sum\_\{i\; =\; 1\}^\{\backslash infty\}\; \backslash tau\; (C\_\{i\})\; \backslash right|\; \backslash begin\{matrix\}\; C\_\{i\}\; \backslash in\; \backslash Sigma,\; \backslash \backslash \; \backslash mathrm\{diam\}\; (C\_\{i\})\; \backslash leq\; \backslash delta,\; \backslash \backslash \; \backslash bigcup\_\{i\; =\; 1\}^\{\backslash infty\}\; C\_\{i\}\; \backslash supseteq\; E\; \backslash end\{matrix\}\; \backslash right\backslash \},$

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 as δ → 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

:$\backslash tau(C)\; =\; \backslash mathrm\{diam\}\; (C)^s,\backslash ,$

where s is a positive constant and where τ is defined on the power set of all subsets of X (i.e., $\backslash Sigma\; =\; 2^X$), 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 balls of X, the associated measure $S^s$ is an s-dimensional spherical measure (not to be confused with the usual spherical measure), the following inequality applies:

$H^s(E)\; \backslash le\; S^s(E)\; \backslash le\; 2H^s(E)$. {{clarify|reason=Covering by balls gives a different result than covering by general shapes; how this works needs to be explained in detail.}}

{{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, $H^d(S)$ 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

:$\backslash frac\{\backslash pi^\{d/2\}\}\{\backslash Gamma(\backslash frac\{d\}\{2\}+1)\}.$https://encyclopediaofmath.org/wiki/Hausdorff_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

:$P\_0^s\; (S)\; =\; \backslash limsup\_\{\backslash delta\; \backslash downarrow\; 0\}\backslash left\backslash \{\; \backslash left.\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash mathrm\{diam\}\; (B\_i)^s\; \backslash right|\; \backslash begin\{matrix\}\; \backslash \{\; B\_i\; \backslash \}\_\{i\; \backslash in\; I\}\; \backslash text\{\; is\; a\; countable\; collection\}\; \backslash \backslash \; \backslash text\{of\; pairwise\; disjoint\; closed\; balls\; with\}\; \backslash \backslash \; \backslash text\{diameters\; \}\; \backslash leq\; \backslash delta\; \backslash text\{\; and\; centres\; in\; \}\; S\; \backslash end\{matrix\}\; \backslash right\backslash \}.$https://projecteuclid.org/download/pdf_1/euclid.rae/1214571371

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

:$P^s\; (S)\; =\; \backslash inf\; \backslash left\backslash \{\; \backslash left.\; \backslash sum\_\{j\; \backslash in\; J\}\; P\_0^s\; (S\_j)\; \backslash right|\; S\; \backslash subseteq\; \backslash bigcup\_\{j\; \backslash in\; J\}\; S\_j,\; J\; \backslash text\{\; countable\}\; \backslash right\backslash \},$

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

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