Hausdorff measure

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

${\displaystyle \operatorname {diam} U:=\sup\{\rho (x,y):x,y\in U\},\quad \operatorname {diam} \emptyset$ :=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 nonincreasing 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 Carathéodory's extension theorem, 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, we can require the covering sets to be open or closed, or in normed spaces even convex, that will yield the same ${\displaystyle H_{\delta }^{d}(S)}$ numbers, hence the same measure. In ${\displaystyle \mathbb {R} ^{n}}$ restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.

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 α_d 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.

It turns out that if ${\displaystyle H^{d}(S)}$ may have a finite, nonzero value for at most one ${\displaystyle d}$. That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to how the area of a line is zero and the length of a 2D shape is infinity. This leads to one of several possible equivalent definitions of the Hausdorff dimension:

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

where we take

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

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.

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 ${\displaystyle m<n}$, 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 ${\displaystyle n=2}$.

• Hausdorff dimension
• Geometric measure theory
• Measure theory
• Outer measure

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• Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4.
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• Morgan, Frank (1988), Geometric Measure Theory, Academic Press.
• Rogers, C. A. (1998), Hausdorff measures, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 0-521-62491-6
• Szpilrajn, E (1937), "La dimension et la mesure" (PDF), Fundamenta Mathematicae, 28: 81–89.

• Hausdorff dimension at Encyclopedia of Mathematics
• Hausdorff measure at Encyclopedia of Mathematics