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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.[1] 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.[2] 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

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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[3]

where

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 μδ(E) increases as δ decreases.)

The function and domain of τ may determine the specific measure obtained. For instance, if we give

where s is a positive constant and where τ is defined on the power set of all subsets of X (i.e., ), 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 is an s-dimensional spherical measure (not to be confused with the usual spherical measure), the following inequality applies:

.[3] [clarification needed]

Hausdorff measure

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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, 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

[4]

Packing measure

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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 (Xd) be a metric space with a subset S ⊆ X and let s ≥ 0. We take a "pre-measure" of S, defined to be

[5]

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

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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