# 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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle (X,\rho )}$ be a metric space. For any subset ${\displaystyle U\subset X}$, let ${\displaystyle \mathrm {diam} \;U}$ denote its diameter, that is

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

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

${\displaystyle H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},}$

where the infimum is over all countable covers of ${\displaystyle S}$ by sets ${\displaystyle U_{i}\subset X}$ satisfying ${\displaystyle \operatorname {diam} U_{i}<\delta }$.

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

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

It can be seen that ${\displaystyle H^{d}(S)}$ 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 ${\displaystyle d}$-dimensional Hausdorff measure of ${\displaystyle S}$. Due to the metric outer measure property, all Borel subsets of ${\displaystyle X}$ are ${\displaystyle H^{d}}$ 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 ${\displaystyle H_{\delta }^{d}(S)}$ may be different (Federer 1969, §2.10.2). If ${\displaystyle X}$ 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.[1]

## Properties of Hausdorff measures

Note that if d is a positive integer, the d dimensional Hausdorff measure of ${\displaystyle \mathbb {R} ^{d}}$ is a rescaling of usual d-dimensional Lebesgue measure ${\displaystyle \lambda _{d}}$ which is normalized so that the Lebesgue measure of the unit cube [0,1]d is 1. In fact, for any Borel set E,

${\displaystyle \lambda _{d}(E)=2^{-d}\alpha _{d}H^{d}(E),}$

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

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

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

${\displaystyle \dim _{\mathrm {Haus} }(S)=\inf\{d\geq 0:H^{d}(S)=0\}=\sup \left(\{d\geq 0:H^{d}(S)=\infty \}\cup \{0\}\right),}$

where we take

${\displaystyle \inf \emptyset =\infty .}$

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

In fractal geometry, some fractals with Hausdorff dimension ${\displaystyle d}$ have zero or infinite ${\displaystyle d}$-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 ${\displaystyle (\operatorname {diam} U_{i})^{d}}$ is replaced with ${\displaystyle \phi (U_{i}),}$ where ${\displaystyle \phi }$ is any monotone increasing set function satisfying ${\displaystyle \phi (\emptyset )=0.}$

This is the Hausdorff measure of ${\displaystyle S}$ with gauge function ${\displaystyle \phi ,}$ or ${\displaystyle \phi }$-Hausdorff measure. A ${\displaystyle d}$-dimensional set ${\displaystyle S}$ may satisfy ${\displaystyle H^{d}(S)=0,}$ but ${\displaystyle H^{\phi }(S)\in (0,\infty )}$ with an appropriate ${\displaystyle \phi .}$ Examples of gauge functions include

${\displaystyle \phi (t)=t^{2}\log \log {\frac {1}{t}}\quad {\text{or}}\quad \phi (t)=t^{2}\log {\frac {1}{t}}\log \log \log {\frac {1}{t}}.}$

The former gives almost surely positive and ${\displaystyle \sigma }$-finite measure to the Brownian path in ${\displaystyle \mathbb {R} ^{n}}$ when ${\displaystyle n>2}$, and the latter when ${\displaystyle n=2}$.