Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rⁿ 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. The one-dimensional Hausdorff measure of a simple curve in Rⁿ is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R² is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes 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 $(X,\rho)$ be a metric space. For any subset $\scriptstyle U\subset X$ , let $\mathrm{diam}\;U$ denote its diameter, that is

\mathrm{diam}\;U :=\sup\{\rho(x,y)|x,y\in U\}, \quad \mathrm{diam}\;\emptyset:=0

Let $S$ be any subset of $X$ , and $\delta>0$ a real number. Define

H^d_\delta(S)=\inf\Bigl\{\sum_{i=1}^\infty (\operatorname{diam}\;U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S,\,\operatorname{diam}\;U_i<\delta\Bigr\}.

(The infimum is over all countable covers of $S$ by sets $\scriptstyle U_i\subset X$ satisfying $\scriptstyle \operatorname{diam}\;U_i<\delta$ .)

Note that $\scriptstyle H^d_\delta(S)$ is monotone decreasing in $\delta$ since the larger $\delta$ is, the more collections of sets are permitted, making the infimum smaller. Thus, the limit $\scriptstyle\lim_{\delta\to 0}H^d_\delta(S)$ exists but may be infinite. Let

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

It can be seen that $H^d(S)$ is an outer measure (more precisely, it is a metric outer measure). By general theory, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the $d$ -dimensional Hausdorff measure of $S$ . Due to the metric outer measure property, all Borel subsets of $X$ are $H^d$ measurable.

In the above definition the sets in the covering are arbitrary. However, they may be taken to be open or closed, and will yield the same measure, although the approximations $\scriptstyle H^d_\delta(S)$ may be different (Federer 1969, §2.10.2). If $X$ is a normed space the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different measure.^[1]

Properties of Hausdorff measures

Note that if d is a positive integer, the d dimensional Hausdorff measure of R^d is a rescaling of usual d-dimensional Lebesgue measure $\lambda_d$ 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} \alpha_d H^d(E)\,

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

\alpha_d =\frac{\Gamma(\frac12)^d}{\Gamma(\frac{d}{2}+1)} =\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}.

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

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. \,

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 $\scriptstyle\mathbb{R}^n$ is said to be $m$ -rectifiable if it is the image of a bounded set in $\scriptstyle\mathbb{R}^n$ under a Lipschitz function. If $m<n$ , then the $m$ -dimensional Minkowski content of a closed $m$ -rectifiable subset of $\scriptstyle\mathbb{R}^n$ is equal to $2^{-m}\alpha_m$ times the $m$ -dimensional Hausdorff measure (Federer 1969, Theorem 3.2.29).

In fractal geometry, some fractals with Hausdorff dimension $d$ have zero or infinite $d$ -dimensional Hausdorff measure. For example, almost surely 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:

In the definition of the measure

|U_i|^d

is replaced with

\phi(U_i)

, where

\phi

is any monotone increasing set function satisfying

\phi(\emptyset )=0

.

This is the Hausdorff measure of $S$ with gauge function $\phi$ , or $\phi$ -Hausdorff measure. A $d$ -dimensional set $S$ may satisfy $H^d(S)=0$ , but $\scriptstyle H^\phi(S)\in(0,\infty)$ with an appropriate $\phi.$ Examples of gauge functions include $\scriptstyle \phi(t)=t^2\,\log\log\frac 1t$ or $\scriptstyle\phi(t) = t^2\log\frac{1}{t}\log\log\log\frac{1}{t}$ . The former gives almost surely positive and $\sigma$ -finite measure to the Brownian path in $\scriptstyle\mathbb{R}^n$ when $n>2$ , and the latter when $n=2$ .