View source for Hausdorff measure

{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

==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.

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 $E$. 

'''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 $\mathcal{H}^\alpha_\delta$ 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.

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 {{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}}).

'''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|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 ther 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
\[
\mu (A) := \mathcal{H}^\alpha (A\cap E) \qquad \mbox{for $A\subset X$ $\mathcal{H}^\alpha$-measurable}
\]
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 of dimension $n$, $\mathcal{H}^n$ coincides with the standard volume form. 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).
|-
|valign="top"|{{Ref|KS}}|| J.-P. Kahane,   R. Salem,   "Ensembles parfaits et séries  trigonométriques" , Hermann  (1963)  pp. 142  {{MR|0160065}}  {{ZBL|0112.29304}}
|-
|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}}
|-
|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}}
|-
|valign="top"|{{Ref|Mu}}|| M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953). {{MR|1528565}} {{MR|0053186}} {{ZBL|0050.05603}}
|-
|valign="top"|{{Ref|Ro}}||  C.A. Rogers,   "Hausdorff measures" , Cambridge Univ. Press  (1970)  {{MR|0281862}} {{ZBL|0204.37601}}
|-
|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}}
|-
|}