Hausdorff measure
2010 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]
Contents
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}\tag{1} \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 (1) 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 (1) 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 (1). 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
- Harmonic analysis (see for instance [KS])
- Potential theory (see for instance [Ca]; Hausdorff measures are closely linked to capacities, cp. with [De]).
- The metric theory of continued fractions (cf. with [Ro]).
- In complex analysis (cp. with Painleve problem and [Ch]).
- In partial differential equations and differential geometry.
References
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Hausdorff measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hausdorff_measure&oldid=30102