Hausdorff measure

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In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.

The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory.

Definition[edit]

Let be a metric space. For any subset , let denote its diameter, that is

Let be any subset of and a real number. Define

where the infimum is over all countable covers of by sets satisfying .

Note that is monotone nonincreasing in since the larger is, the more collections of sets are permitted, making the infimum not larger. Thus, exists but may be infinite. Let

It can be seen that is an outer measure (more precisely, it is a metric outer measure). By Carathéodory's extension theorem, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the -dimensional Hausdorff measure of . Due to the metric outer measure property, all Borel subsets of are measurable.

In the above definition the sets in the covering are arbitrary.

However, we can require the covering sets to be open or closed, or in normed spaces even convex, that will yield the same numbers, hence the same measure. In restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.

Properties of Hausdorff measures[edit]

Note that if d is a positive integer, the d dimensional Hausdorff measure of is a rescaling of usual d-dimensional Lebesgue measure which is normalized so that the Lebesgue measure of the unit cube [0,1]d is 1. In fact, for any Borel set E,

where αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function

Remark. Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that it is normalized in such a way that Hausdorff d-dimensional measure in the case of Euclidean space coincides exactly with Lebesgue measure.

Relation with Hausdorff dimension[edit]

It turns out that may have a finite, nonzero value for at most one . That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to the idea that the area of a line is zero and the length of a 2D shape is in some sense infinity. This leads to one of several possible equivalent definitions of the Hausdorff dimension:

where we take

Note that it is not guaranteed that the Hausdorff measure must be finite and nonzero for some d, and indeed the measure at the Hausdorff dimension may still be zero; in this case, the Hausdorff dimension still acts as an inflection point between measures of zero and infinity.

Generalizations[edit]

In geometric measure theory and related fields, the Minkowski content is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of is said to be -rectifiable if it is the image of a bounded set in under a Lipschitz function. If , then the -dimensional Minkowski content of a closed -rectifiable subset of is equal to times the -dimensional Hausdorff measure (Federer 1969, Theorem 3.2.29).

In fractal geometry, some fractals with Hausdorff dimension have zero or infinite -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdorff measure is zero. In order to “measure” the “size” of such sets, mathematicians have considered the following variation on the notion of the Hausdorff measure:

In the definition of the measure is replaced with where is any monotone increasing set function satisfying

This is the Hausdorff measure of with gauge function or -Hausdorff measure. A -dimensional set may satisfy but with an appropriate Examples of gauge functions include

The former gives almost surely positive and -finite measure to the Brownian path in when , and the latter when .

See also[edit]

References[edit]

  • Evans, Lawrence C.; Gariepy, Ronald F. (1992), Measure Theory and Fine Properties of Functions, CRC Press.
  • Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4.
  • Hausdorff, Felix (1918), "Dimension und äusseres Mass" (PDF), Mathematische Annalen, 79 (1–2): 157–179, doi:10.1007/BF01457179.
  • Morgan, Frank (1988), Geometric Measure Theory, Academic Press.
  • Rogers, C. A. (1998), Hausdorff measures, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 0-521-62491-6
  • Szpilrajn, E (1937), "La dimension et la mesure" (PDF), Fundamenta Mathematicae, 28: 81–89, doi:10.4064/fm-28-1-81-89.

External links[edit]