Hausdorff measure

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In mathematics a Hausdorff measure assigns a number in [0,\infty] to every metric space. The zero dimensional Hausdorff measure of a metric space is the number of points in the space (if the space is finite) or \infty if the space is infinite. The one dimensional Hausdorff measure of a metric space which is an imbedded path in \R^n is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d\ge 0 which is not necessarily an integer. These measures are useful for studying the size of fractals.

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[edit] Definition

Fix some d\ge 0 and a metric space X. Let S\subset X be any subset of X. For δ > 0 let

H^d_\delta(S):=\inf\Bigl\{\sum_i r_i^d : \text{there is a collection of balls with radii }r_i\in(0,\delta)\text{ which cover }S\Bigr\}.

Note that H^d_\delta(S) is monotone decreasing in δ since the larger δ is, the more collections of balls are permitted. Thus, the limit \lim_{\delta\to 0}H^d_\delta(S) exists. Set

H^d(S):=\sup_{\delta>0} H^d_\delta(S)=\lim_{\delta\to 0}H^d_\delta(S).

This is the d-dimensional Hausdorff measure of S.

[edit] Properties of Hausdorff measures

The Hausdorff measures Hd are outer measures. Moreover, all Borel subsets of X are Hd measureable. In particular, the theory of outer measures implies that Hd is countably additive on the Borel σ-field.

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

\lambda_d(E) = 2^{-d} \frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)} H^d(E).

Remark. Some authors adopt a slightly different definition of Hausdorff measure than 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.

[edit] Relation with Hausdorff dimension

One of several possible equivalent definitions of the Hausdorff dimension is

\operatorname{dim}_{\mathrm{Haus}}(S):= \inf\{d\ge 0:H^d(S)=0\}=\sup\bigl(\{d\ge 0:H^d(S)=\infty\}\cup\{0\}\bigr),

where we take \inf\emptyset=\infty.

[edit] Generalizations

Some fractals with Hausdorff dimension d have zero or infinite d-dimensional Hausdorff measure. For example, the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdoff 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. Consider any monotone increasing function \phi:[0,\infty)\to[0,\infty) satisfying φ(0) = 0. Set

H^{\phi}_\delta(S):=\inf\Bigl\{\sum_i \phi(r_i) : \text{there is a collection of balls with radii }r_i\in(0,\delta)\text{ which cover }S\Bigr\},

and

H^\phi(S):=\sup_{\delta>0} H^\phi_\delta(S)=\lim_{\delta\to 0}H^\phi_\delta(S).

This is the Hausdorff measure of S with gauge function φ. For example, a d-dimensional fractal S may satisfy Hd(S) = 0, but H^\phi(S)\in(0,\infty), where \phi(r)=r^d\,\log(2+1/r).

[edit] References