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 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.

A Carathéodory construction is a constructive method of building fractal measures, used to create measures from similarly defined outer measures.

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


 * $$\mu (E) = \lim_{\delta \to 0} \mu_{\delta} (E),$$

where
 * $$\mu_{\delta} (E) = \inf \left\{ \left. \sum_{i = 1}^{\infty} \tau (C_{i}) \right| \begin{matrix} C_{i} \in \Sigma, \\ \mathrm{diam} (C_{i}) \leq \delta, \\   \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \end{matrix} \right\},$$

is not only an outer measure, but in fact a metric outer measure as well. (Some authors prefer to take a supremum over δ &gt; 0 rather than a limit as δ → 0; the two give the same result, since μδ(E) increases as δ decreases.)

The function and domain of τ may determine the specific measure obtained. For instance, if we give
 * $$\tau(C) = \mathrm{diam} (C)^s,\,$$

where s is a positive constant and where τ is defined on the power set of all subsets of X (i.e., $$\Sigma = 2^X$$), 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:

$$H^s(E) \le S^s(E) \le 2H^s(E)$$.

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

Packing measure
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
 * $$P_0^s (S) = \limsup_{\delta \downarrow 0}\left\{ \left. \sum_{i \in I} \mathrm{diam} (B_i)^s \right| \begin{matrix} \{ B_i \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint closed balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}.$$

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.