Hausdorff measure

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 formula_1 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 formula_1 is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of formula_3 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.
Let formula_4 be a metric space. For any subset formula_5, let formula_6 denote its diameter, that is
Let formula_8 be any subset of formula_9 and formula_10 a real number. Define
where the infimum is over all countable covers of formula_8 by sets formula_13 satisfying formula_14.
Note that formula_15 is monotone nonincreasing in formula_16 since the larger formula_17 is, the more collections of sets are permitted, making the infimum not larger. Thus, formula_18 exists but may be infinite. Let
It can be seen that formula_20 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 formula_21-dimensional Hausdorff measure of formula_8. Due to the metric outer measure property, all Borel subsets of formula_23 are formula_24 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 formula_15 numbers, hence the same measure. In formula_1 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.
Note that if "d" is a positive integer, the "d" dimensional Hausdorff measure of formula_27 is a rescaling of usual "d"-dimensional Lebesgue measure formula_28 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 α"d" 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.
It turns out that if formula_20 may have a finite, nonzero value for at most one formula_21. 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.
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 formula_1 is said to be formula_36-rectifiable if it is the image of a bounded set in formula_37 under a Lipschitz function. If formula_38, then the formula_36-dimensional Minkowski content of a closed formula_36-rectifiable subset of formula_1 is equal to formula_42 times the formula_36-dimensional Hausdorff measure .
In fractal geometry, some fractals with Hausdorff dimension formula_21 have zero or infinite formula_21-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:
This is the Hausdorff measure of formula_8 with gauge function formula_51 or formula_48-Hausdorff measure. A formula_21-dimensional set formula_8 may satisfy formula_55 but formula_56 with an appropriate formula_57 Examples of gauge functions include
The former gives almost surely positive and formula_59-finite measure to the Brownian path in formula_1 when formula_61, and the latter when formula_62.