# Hausdorff measure

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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 $\mathbb {R} ^{n}$ 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 $\mathbb {R} ^{n}$ is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of $\mathbb {R} ^{2}$ 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 $U\subset X$ , 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

$H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},$ where the infimum is over all countable covers of $S$ by sets $U_{i}\subset X$ satisfying $\operatorname {diam} U_{i}<\delta$ .

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

$H^{d}(S):=\sup _{\delta >0}H_{\delta }^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S).$ 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 $H_{\delta }^{d}(S)$ 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, yet comparable, measure.

## Properties of Hausdorff measures

Note that if d is a positive integer, the d dimensional Hausdorff measure of $\mathbb {R} ^{d}$ 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 αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function

$\alpha _{d}={\frac {\Gamma ({\frac {1}{2}})^{d}}{\Gamma ({\frac {d}{2}}+1)}}={\frac {\pi ^{d/2}}{\Gamma ({\frac {d}{2}}+1)}}.$ 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

$\dim _{\mathrm {Haus} }(S)=\inf\{d\geq 0:H^{d}(S)=0\}=\sup \left(\{d\geq 0:H^{d}(S)=\infty \}\cup \{0\}\right),$ 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 $\mathbb {R} ^{n}$ is said to be $m$ -rectifiable if it is the image of a bounded set in $\mathbb {R} ^{m}$ under a Lipschitz function. If $m , then the $m$ -dimensional Minkowski content of a closed $m$ -rectifiable subset of $\mathbb {R} ^{n}$ 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 $(\operatorname {diam} U_{i})^{d}$ 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 $H^{\phi }(S)\in (0,\infty )$ with an appropriate $\phi .$ Examples of gauge functions include

$\phi (t)=t^{2}\log \log {\frac {1}{t}}\quad {\text{or}}\quad \phi (t)=t^{2}\log {\frac {1}{t}}\log \log \log {\frac {1}{t}}.$ The former gives almost surely positive and $\sigma$ -finite measure to the Brownian path in $\mathbb {R} ^{n}$ when $n>2$ , and the latter when $n=2$ .