Hausdorff measure

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 \[ \mu (A) := \mathcal{H}^\alpha (A\cap E) \qquad \mbox{for '"`UNIQ-MathJax49-QINU`"' '"`UNIQ-MathJax50-QINU`"'-measurable} \] 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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