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A collective name for the class of measures determined on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467001.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467002.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467003.png" /> by means of the following construction: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467004.png" /> be a certain class of open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467005.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467006.png" /> be a non-negative function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467007.png" />, and let
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{{MSC|28A}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467008.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467009.png" /></td> </tr></table>
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{{TEX|done}}
  
where the infimum is taken over all finite or countable coverings of the Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670010.png" /> by sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670011.png" /> with diameter not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670012.png" />. The Hausdorff measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670013.png" /> defined by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670014.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670015.png" /> is the limit
+
==Definition==
 +
The term Hausdorff measures is used for a class of [[Outer measure|outer measures]] (introduced for the first time by Hausdorff in {{Cite|Ha}}) on subsets of a generic metric space $(X,d)$, or for their restrictions to the corresponding measurable sets.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670016.png" /></td> </tr></table>
+
Let $(X,d)$ be a metric space. In what follows, for any subset  $E\subset X$, ${\rm diam}\, (E)$ will denote the [[Diameter|diameter]]  of $E$.  
  
Examples of Hausdorff measures. 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670017.png" /> be the collection of all balls in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670018.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670020.png" />. The corresponding measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670021.png" /> is called the Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670023.png" />-measure (the linear Hausdorff measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670024.png" /> and the plane Hausdorff measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670025.png" />). 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670026.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670027.png" /> be the collection of cylinders with spherical bases and with axes parallel to the direction of the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670028.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670029.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670030.png" />-dimensional volume of the axial section of a cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670031.png" />; the corresponding Hausdorff measure is called the cylindrical measure.
+
'''Definition 1'''
 +
For any $E\subset X$,  any $\delta \in ]0, \infty]$ and any $\alpha\in [0, \infty[$ we  consider the [[Outer measure|outer measure]]
 +
\begin{equation}\label{e:hausdorff_m}
 +
\mathcal{H}^\alpha_\delta (E) :=  
 +
\omega_\alpha \inf \left\{ \sum_{i=1}^\infty ({\rm diam}\, E_i)^\alpha :
 +
E\subset \bigcup_i E_i \quad\mbox{and}\quad {\rm diam}\, (E_i) < \delta\right\}\, ,
 +
\end{equation}
 +
where $\omega_\alpha$ is a positive factor (see below for the precise definition).
  
The Hausdorff measures were introduced by F. Hausdorff [[#References|[1]]].
+
The $\mathcal{H}^\alpha_\delta$ defined above are [[Outer measure|outer measures]] and they are called ''Hausdorff premeasures'' by some authors. Moreover, in \eqref{e:hausdorff_m} the infimum can be taken over open coverings or closed coverings without changing the result.
  
====References====
+
The  map $\delta\mapsto \mathcal{H}^\alpha_\delta (E)$ is monotone nonincreasing and thus we can define the ''Hausdorff $\alpha$-dimensional measure'' (or ''Hausdorff $\alpha$-dimensional outer measure'') of $E$ as
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Hausdorff,  "Dimension and äusseres Mass"  ''Math. Ann.'' , '''79'''  (1918) pp. 157–179</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
+
\[
 +
\mathcal{H}^\alpha (E) := \lim_{\delta\downarrow 0} \mathcal{H}^\alpha_\delta (E)\, .
 +
\]
  
 +
'''Remark 2'''
 +
The normalization constant $\omega_\alpha$ is equal to
 +
\[
 +
\omega_\alpha = \frac{\pi^{\alpha/2}}{\Gamma \left(\frac{\alpha}{2}+1\right)}\,
 +
\]
 +
(cp. with Section 2.1 of {{Cite|EG}}).
 +
When $\alpha$ is a (positive) integer $n$, $\omega_n$ equals the Lebesgue measure of the unit ball in $\mathbb R^n$. With
 +
this choice the $n$-dimensional Hausdorff outer measure on the euclidean space $\mathbb R^n$ coincides
 +
with the Lebesgue measure. However some authors set $\omega_\alpha =1$ (see for instance {{Cite|Ma}}).
  
 +
===Hausdorff dimension===
 +
The following is a simple consequence of the definition (cp. with Theorem 4.7 of {{Cite|Ma}}).
  
====Comments====
+
'''Theorem 3'''
A method to construct measures on metric spaces was introduced by C. Carathéodory in 1914. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670032.png" /> can be anything and are often taken closed. The Hausdorff measures are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670033.png" />-additive on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670034.png" />-field, but are in general not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670035.png" />-finite; some restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670038.png" /> have to be added in order to get nice properties of approximation from below. Such restrictions are, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670039.png" /> is a Borel subset of some compact metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670040.png" /> is the class of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670042.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670043.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670044.png" /> is a continuous non-decreasing function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670046.png" />. The measures obtained in this way are the Hausdorff measures most often used, and were mainly studied by A.S. Besicovitch and his school (cf. [[#References|[a8]]]); they are called (Hausdorff) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670047.png" />-measures (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670049.png" />, one says <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670050.png" />-measure or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670051.png" />-dimensional measure; see also [[Hausdorff dimension|Hausdorff dimension]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670052.png" /> is the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670053.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670054.png" />-dimensional measure is equal (up to a multiplicative constant) to the [[Lebesgue measure|Lebesgue measure]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670055.png" /> and, when restricted to smooth curves, surfaces, etc., is equal to the length, area, etc. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670056.png" />, etc. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670057.png" />-measure is the counting measure, which is also at the confines of potential theory and descriptive set theory.
+
For $0\leq s<t<\infty$ and $A\subset X$ we have
 +
* $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
 +
* $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.
  
Even though the notion of Hausdorff measure is not regarded as being fundamental, it appears in many parts of hard analysis and geometry. Through the study of exceptional sets, it is used, for example, in [[Harmonic analysis|harmonic analysis]] (cf. [[#References|[a5]]]), in [[Potential theory|potential theory]] (cf. [[#References|[a1]]]), in the metric theory of continued fractions (cf. [[#References|[a8]]]), and in differential geometry (cf. [[Sard theorem|Sard theorem]]). For many reasons it is closely linked to the notion of [[Capacity|capacity]], and, more generally, to [[Descriptive set theory|descriptive set theory]] (cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a8]]]; R.O. Davies was, before G. Choquet, the first to prove capacitability theorems). In the theory of stochastic processes it has a crucial role in the fine study of the paths of the [[Wiener process|Wiener process]] and others (cf. [[#References|[a6]]] and its bibliography). Finally, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670058.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670059.png" />-dimensional measure is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670060.png" /> is a natural integer, a fundamental notion in geometric measure theory (cf. [[Area|Area]]; [[Minimal surface|Minimal surface]], and [[#References|[a4]]] in which also Hausdorff measures which are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670061.png" />-measures ([[Favard measure|Favard measure]], etc.) are used); for non-integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670062.png" /> it is a fundamental notion in the theory of [[Fractals|fractals]] (cf. [[#References|[a3]]]).
+
The [[Hausdorff dimension]] ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as
  
For the use of Hausdorff measures and the [[Hausdorff dimension|Hausdorff dimension]] in multi-dimensional complex analysis, see, e.g., [[#References|[a9]]].
+
'''Definition 4'''
 +
\begin{align*}
 +
{\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\
 +
&=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, .
 +
\end{align*}
  
====References====
+
===Generalizations===
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"C. Dellacherie,  "Ensembles analytiques, capacités, mesures de Hausdorff" , ''Lect. notes in math.'' , '''295''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K.J. Falconer,  "The geometry of fractal sets" , Cambridge Univ. Press (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Federer,  "Geometric measure theory" , Springer (1969)  pp. 60; 62; 71; 108</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"J.-P. Kahane,  R. Salem,  "Ensembles parfaits et séries trigonométriques" , Hermann  (1963)  pp. 142</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.-F. le Gall,  "Temps locaux d'intersection et points multiples des processus de Lévy" , ''Sem. Probab. XXI'' , ''Lect. notes in math.'' , '''1247''' , Springer  (1987)  pp. 341–374</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"M.E. Munroe,   "Introduction to measure and integration" , Addison-Wesley (1953) pp. 111</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> C.A. Rogers,  "Hausdorff measures" , Cambridge Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989) (Translated from Russian)</TD></TR></table>
+
The definition of the Hausdorff measures is just a special case of a more general construction due to Caratheodory, which starting from a generic (nonnegative) [[Set function|set function]] $\nu$ with $\nu (\emptyset) =0$ builds an outer measure $\mu$ (we refer to [[Outer measure]] for a decription of Caratheodory's method). A generalization of the usual Hausdorff measures replaces $\omega_\alpha ({\rm diam}\, (E_i))^\alpha$ in \ref{e:hausdorff_m} with $h ({\rm diam}\, (E_i))$, where $h: \mathbb R^+\to \mathbb R^+$ is a nondecreasing function (often called ''gauge function''). See for instance {{Cite|Ma}}.
 +
 
 +
The construction of Caratheodory allows for several other outer measures in the Euclidean space, most of which coincide with the Hausdorff $k$-dimensional measures for $C^1$ submanifolds when $k$ is an integer, but differ on general sets. One example is the [[Favard measure]], also called integralgeometric measure. See {{Cite|Fe}} and {{Cite|KP}}.
 +
 
 +
In some common generalizations of the Hausdorff measures one restricts the class of admissible coverings in \ref{e:hausdorff_m}. For instance one can use coverings by balls (and the resulting outer measure is then called ''spherical Hausdorff measure'') or by cylinders (''cylindrical Hausdorff measure'').
 +
 
 +
==Measure-theoretic properties==
 +
The Hausdorff measures $\mathcal{H}^\alpha$ satisfy [[Caratheodory criterion|Caratheodory's criterion]]. Therefore, the [[Algebra of sets|$\sigma$-algebra]] of $\mathcal{H}^\alpha$-measurable sets (see [[Outer measure]] for the definition) contains the [[Borel set|Borel sets]] (i.e. $\mathcal{H}^\alpha$ is a [[Borel measure|Borel outer measure]]). The Hausdorff measures are
 +
also ''Borel regular'', in the sense that, for any set $A\subset X$ there is a Borel set $B\supset A$ with $\mathcal{H}^\alpha (B) = \mathcal{H}^\alpha (A)$ (see Corollary 4.5 in {{Cite|Ma}}).
 +
 
 +
'''Remark 5'''
 +
The premeasures $\mathcal{H}^\alpha_\delta$ do not satisfy Caratheodory's criterion and, moreover, they are not necessarily Borel outer measures: this property fails already in the Euclidean spaces (see {{Cite|Si}}).
 +
 
 +
If $E\subset X$ is $\mathcal{H}^\alpha$ measurable and $\mathcal{H}^\alpha (E)<\infty$, then the measure
 +
\[
 +
\mu (A) := \mathcal{H}^\alpha (A\cap E) \qquad \mbox{for } A\subset X \;\, \mathcal{H}^\alpha\text{-measurable}
 +
\]
 +
is a [[Radon measure]] (see p. 57 of {{Cite|Ma}}).
 +
 
 +
==$\mathcal{H}^n$ for $n$ integer==
 +
For $n$ integer the Hausdorff measures are suitable measure-theoretic generalizations of the concept of $n$-dimensional volume of a smooth Riemannian manifold.
 +
 
 +
===The counting measure===
 +
In any metric space $(X,d)$ and for any set $E\subset X$, $\mathcal{H}^0 (E)$ equals the cardinality of $E$ if $E$ is a finite set and it equals infinity if not. $\mathcal{H}^0$ is called, therefore, the ''counting measure''.
 +
 
 +
===Length===
 +
In any metric space $(X,d)$, if $\gamma: [0,1]\to X$ is an injective Lipschitz function, then $\mathcal{H}^1 (\gamma ([0,1])$ is the length of the curve (see [[Rectifiable curve]] for the relevant definition).
 +
 
 +
===$n$-dimensional volume===
 +
On the euclidean space $\mathbb R^n$ $\mathcal{H}^n$ coincides with the [[Lebesgue measure|Lebesgue outer measure]] (see Theorem 2 in Section 2.2 of {{Cite|EG}}). More generally, in a Riemannian manifold $M$ of dimension $n$, $\mathcal{H}^n$ coincides with the standard volume. Thus, If $\Sigma$ is a $C^1$ submanifold of $\mathbb R^N$ of dimension $n$, then $\mathcal{H}^n (\Gamma)$ is the usual $n$-dimensional volume of $\Gamma$. In this case a useful tool to compute the Hausdorff measure is the [[Area formula]].
 +
 
 +
====Rectifiable sets====
 +
For several applications, the class of Borel sets of $\mathbb R^N$ with finite $\mathcal{H}^n$ measure is too large to be an appropriate generalization of smooth $n$-dimensional surfaces. An intermediate class which has wide applications is that of [[Rectifiable set|rectifiable sets]].
 +
 
 +
==Relations to density==
 +
Especially in the euclidean space there is a strong link between various concepts of ''densities'' of measures and sets and the Hausdorff measures (see [[Density of a set]]). This relation, pioneered by Besicovitch and his school (cf. {{Cite|Ro}}), plays a fundamental role in [[Geometric measure theory]] (see for instance {{Cite|Fe}}, {{Cite|KP}} or {{Cite|Si}}).
 +
 
 +
==Relevance==
 +
Hausdorff measures play an important role in several areas of mathematics
 +
* They are fundamental in [[Geometric measure theory]], especially in the solution of the Plateau problem (see also [[Minimal surface]]).
 +
* They are a fundamental notion in the theory of fractals, see {{Cite|Fa}}.
 +
* In the theory of stochastic processes they have a crucial role  in the fine study of the paths of the [[Wiener process|Wiener process]]  and others (cf. {{Cite|lG}}).
 +
 
 +
Through the study of exceptional sets they are widely used in
 +
* [[Harmonic analysis]] (see for instance {{Cite|KS}})
 +
* [[Potential theory]] (see for instance {{Cite|Ca}}; Hausdorff measures are closely linked to [[Capacity|capacities]], cp. with {{Cite|De}}).
 +
* The metric theory of continued fractions (cf. with {{Cite|Ro}}).
 +
* In complex analysis (cp. with [[Painleve problem]] and {{Cite|Ch}}).
 +
* In partial differential equations and differential geometry.
 +
 
 +
==References==
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Ca}}|| L. Carleson,  "Selected problems on exceptional sets" , v. Nostrand  (1967) {{MR|0225986}} {{ZBL|0189.10903}}
 +
|-
 +
|valign="top"|{{Ref|Ch}}|| E.M. Chirka,  "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)  {{MR|1111477}} {{ZBL|0683.32002}}
 +
|-
 +
|valign="top"|{{Ref|De}}|| C. Dellacherie,  "Ensembles analytiques, capacités, mesures de Hausdorff" , ''Lect. notes in math.'' , '''295''' , Springer  (1972) {{MR|0492152}} {{ZBL|0259.31001}}
 +
|-
 +
|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of    functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,      1992. {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Fa}}|| K.J. Falconer,  "The geometry of fractal sets" , Cambridge Univ. Press   (1985) {{MR|0867284}} {{ZBL|0587.28004}}
 +
|-
 +
|valign="top"|{{Ref|Fe}}||      H. Federer, "Geometric measure  theory". Volume 153 of Die    Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New    York Inc., New  York, 1969.  {{MR|0257325}} {{ZBL|0874.49001}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| F. Hausdorff,  "Dimension and äusseres Mass" ''Math. Ann.'' , '''79''' (1918)  pp. 157–179  {{MR|1511917}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|HW}}|| W. Hurevicz,  G. Wallman,  "Dimension theory" , Princeton Univ. Press  (1948)  {{MR|}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|KP}}|| S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
 +
|-
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|valign="top"|{{Ref|KS}}|| J.-P. Kahane,  R. Salem,  "Ensembles parfaits et séries trigonométriques" , Hermann  (1963)  pp. 142 {{MR|0160065}}  {{ZBL|0112.29304}}
 +
|-
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|valign="top"|{{Ref|lG}}|| J.-F. le Gall,  "Temps locaux d'intersection et points multiples des processus de Lévy" , ''Sem. Probab. XXI'' , ''Lect. notes in math.'' , '''1247''' , Springer  (1987)  pp. 341–374 {{ZBL|0621.60077}}
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|-
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|valign="top"|{{Ref|Ma}}||        P. Mattila, "Geometry of sets  and measures in Euclidean spaces.    Fractals and rectifiability".    Cambridge Studies in Advanced    Mathematics, 44. Cambridge University      Press, Cambridge,  1995.    {{MR|1333890}} {{ZBL|0911.28005}}
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|-
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|valign="top"|{{Ref|Mu}}|| M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953). {{MR|1528565}} {{MR|0053186}} {{ZBL|0050.05603}}
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|-
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|valign="top"|{{Ref|Ro}}|| C.A. Rogers,  "Hausdorff measures" , Cambridge Univ. Press  (1970) {{MR|0281862}} {{ZBL|0204.37601}}
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|-
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|valign="top"|{{Ref|Si}}|| L. Simon, "Lectures on geometric measure theory" , ''Proc. Centre Math. Anal. Austral. National Univ.'' , Centre Math. Anal. 3 Austral. National Univ., Canberra (1983) {{MR|0756417}} {{ZBL|0546.49019}}
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Latest revision as of 09:43, 16 August 2013

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

Definition

The term Hausdorff measures is used for a class of outer measures (introduced for the first time by Hausdorff in [Ha]) on subsets of a generic metric space $(X,d)$, or for their restrictions to the corresponding measurable sets.

Let $(X,d)$ be a metric space. In what follows, for any subset $E\subset X$, ${\rm diam}\, (E)$ will denote the diameter of $E$.

Definition 1 For any $E\subset X$, any $\delta \in ]0, \infty]$ and any $\alpha\in [0, \infty[$ we consider the outer measure \begin{equation}\label{e:hausdorff_m} \mathcal{H}^\alpha_\delta (E) := \omega_\alpha \inf \left\{ \sum_{i=1}^\infty ({\rm diam}\, E_i)^\alpha : E\subset \bigcup_i E_i \quad\mbox{and}\quad {\rm diam}\, (E_i) < \delta\right\}\, , \end{equation} where $\omega_\alpha$ is a positive factor (see below for the precise definition).

The $\mathcal{H}^\alpha_\delta$ defined above are outer measures and they are called Hausdorff premeasures by some authors. Moreover, in \eqref{e:hausdorff_m} the infimum can be taken over open coverings or closed coverings without changing the result.

The map $\delta\mapsto \mathcal{H}^\alpha_\delta (E)$ is monotone nonincreasing and thus we can define the Hausdorff $\alpha$-dimensional measure (or Hausdorff $\alpha$-dimensional outer measure) of $E$ as \[ \mathcal{H}^\alpha (E) := \lim_{\delta\downarrow 0} \mathcal{H}^\alpha_\delta (E)\, . \]

Remark 2 The normalization constant $\omega_\alpha$ is equal to \[ \omega_\alpha = \frac{\pi^{\alpha/2}}{\Gamma \left(\frac{\alpha}{2}+1\right)}\, \] (cp. with Section 2.1 of [EG]). When $\alpha$ is a (positive) integer $n$, $\omega_n$ equals the Lebesgue measure of the unit ball in $\mathbb R^n$. With this choice the $n$-dimensional Hausdorff outer measure on the euclidean space $\mathbb R^n$ coincides with the Lebesgue measure. However some authors set $\omega_\alpha =1$ (see for instance [Ma]).

Hausdorff dimension

The following is a simple consequence of the definition (cp. with Theorem 4.7 of [Ma]).

Theorem 3 For $0\leq s<t<\infty$ and $A\subset X$ we have

  • $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
  • $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.

The Hausdorff dimension ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as

Definition 4 \begin{align*} {\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\ &=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, . \end{align*}

Generalizations

The definition of the Hausdorff measures is just a special case of a more general construction due to Caratheodory, which starting from a generic (nonnegative) set function $\nu$ with $\nu (\emptyset) =0$ builds an outer measure $\mu$ (we refer to Outer measure for a decription of Caratheodory's method). A generalization of the usual Hausdorff measures replaces $\omega_\alpha ({\rm diam}\, (E_i))^\alpha$ in \ref{e:hausdorff_m} with $h ({\rm diam}\, (E_i))$, where $h: \mathbb R^+\to \mathbb R^+$ is a nondecreasing function (often called gauge function). See for instance [Ma].

The construction of Caratheodory allows for several other outer measures in the Euclidean space, most of which coincide with the Hausdorff $k$-dimensional measures for $C^1$ submanifolds when $k$ is an integer, but differ on general sets. One example is the Favard measure, also called integralgeometric measure. See [Fe] and [KP].

In some common generalizations of the Hausdorff measures one restricts the class of admissible coverings in \ref{e:hausdorff_m}. For instance one can use coverings by balls (and the resulting outer measure is then called spherical Hausdorff measure) or by cylinders (cylindrical Hausdorff measure).

Measure-theoretic properties

The Hausdorff measures $\mathcal{H}^\alpha$ satisfy Caratheodory's criterion. Therefore, the $\sigma$-algebra of $\mathcal{H}^\alpha$-measurable sets (see Outer measure for the definition) contains the Borel sets (i.e. $\mathcal{H}^\alpha$ is a Borel outer measure). The Hausdorff measures are also Borel regular, in the sense that, for any set $A\subset X$ there is a Borel set $B\supset A$ with $\mathcal{H}^\alpha (B) = \mathcal{H}^\alpha (A)$ (see Corollary 4.5 in [Ma]).

Remark 5 The premeasures $\mathcal{H}^\alpha_\delta$ do not satisfy Caratheodory's criterion and, moreover, they are not necessarily Borel outer measures: this property fails already in the Euclidean spaces (see [Si]).

If $E\subset X$ is $\mathcal{H}^\alpha$ measurable and $\mathcal{H}^\alpha (E)<\infty$, then the measure \[ \mu (A) := \mathcal{H}^\alpha (A\cap E) \qquad \mbox{for } A\subset X \;\, \mathcal{H}^\alpha\text{-measurable} \] is a Radon measure (see p. 57 of [Ma]).

$\mathcal{H}^n$ for $n$ integer

For $n$ integer the Hausdorff measures are suitable measure-theoretic generalizations of the concept of $n$-dimensional volume of a smooth Riemannian manifold.

The counting measure

In any metric space $(X,d)$ and for any set $E\subset X$, $\mathcal{H}^0 (E)$ equals the cardinality of $E$ if $E$ is a finite set and it equals infinity if not. $\mathcal{H}^0$ is called, therefore, the counting measure.

Length

In any metric space $(X,d)$, if $\gamma: [0,1]\to X$ is an injective Lipschitz function, then $\mathcal{H}^1 (\gamma ([0,1])$ is the length of the curve (see Rectifiable curve for the relevant definition).

$n$-dimensional volume

On the euclidean space $\mathbb R^n$ $\mathcal{H}^n$ coincides with the Lebesgue outer measure (see Theorem 2 in Section 2.2 of [EG]). More generally, in a Riemannian manifold $M$ of dimension $n$, $\mathcal{H}^n$ coincides with the standard volume. Thus, If $\Sigma$ is a $C^1$ submanifold of $\mathbb R^N$ of dimension $n$, then $\mathcal{H}^n (\Gamma)$ is the usual $n$-dimensional volume of $\Gamma$. In this case a useful tool to compute the Hausdorff measure is the Area formula.

Rectifiable sets

For several applications, the class of Borel sets of $\mathbb R^N$ with finite $\mathcal{H}^n$ measure is too large to be an appropriate generalization of smooth $n$-dimensional surfaces. An intermediate class which has wide applications is that of rectifiable sets.

Relations to density

Especially in the euclidean space there is a strong link between various concepts of densities of measures and sets and the Hausdorff measures (see Density of a set). This relation, pioneered by Besicovitch and his school (cf. [Ro]), plays a fundamental role in Geometric measure theory (see for instance [Fe], [KP] or [Si]).

Relevance

Hausdorff measures play an important role in several areas of mathematics

  • They are fundamental in Geometric measure theory, especially in the solution of the Plateau problem (see also Minimal surface).
  • They are a fundamental notion in the theory of fractals, see [Fa].
  • In the theory of stochastic processes they have a crucial role in the fine study of the paths of the Wiener process and others (cf. [lG]).

Through the study of exceptional sets they are widely used in

References

[Ca] L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) MR0225986 Zbl 0189.10903
[Ch] E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002
[De] C. Dellacherie, "Ensembles analytiques, capacités, mesures de Hausdorff" , Lect. notes in math. , 295 , Springer (1972) MR0492152 Zbl 0259.31001
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fa] K.J. Falconer, "The geometry of fractal sets" , Cambridge Univ. Press (1985) MR0867284 Zbl 0587.28004
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha] F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
[HW] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
[KP] S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[KS] J.-P. Kahane, R. Salem, "Ensembles parfaits et séries trigonométriques" , Hermann (1963) pp. 142 MR0160065 Zbl 0112.29304
[lG] J.-F. le Gall, "Temps locaux d'intersection et points multiples des processus de Lévy" , Sem. Probab. XXI , Lect. notes in math. , 1247 , Springer (1987) pp. 341–374 Zbl 0621.60077
[Ma] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Mu] M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953). MR1528565 MR0053186 Zbl 0050.05603
[Ro] C.A. Rogers, "Hausdorff measures" , Cambridge Univ. Press (1970) MR0281862 Zbl 0204.37601
[Si] L. Simon, "Lectures on geometric measure theory" , Proc. Centre Math. Anal. Austral. National Univ. , Centre Math. Anal. 3 Austral. National Univ., Canberra (1983) MR0756417 Zbl 0546.49019
How to Cite This Entry:
Hausdorff measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_measure&oldid=18209
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article