2010 Mathematics Subject Classification: Primary: 28A Secondary: 37F35 [MSN][ZBL]

A numerical invariant of a metric space, introduced by F. Hausdorff in [Ha].

## Contents

* 1 Definition
* 2 Remarks
* 3 Properties
* 4 Other definitions of dimension
* 5 References

### Definition[edit]

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) := \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}

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

Warning Several authors define $\mathcal{H}^\alpha_\delta$ in a way which differs from \ref{e:hausdorff_m} by a multiplicative positive factor $\omega_\alpha$. This factor ensures that $\mathcal{H}^n$ coincides with the Lebesgue (outer) measure when $X$ is the $n$-dimensional euclidean space. In any case the multiplicative factor does not make a difference in the definition of the Hausdorff dimension (see below).

Indeed $\mathcal{H}^\alpha$ is an outer measure and the procedure above is a classical construction (sometimes called Caratheodory construction, see again Outer measure). The following is a simple consequence of the definition (cp. with Theorem 4.7 of [Ma]).

Theorem 2 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 3 \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*}

### Remarks[edit]

In the early developments of Geometric measure theory several seminal papers by Besicovitch played a fundamental role in clarifying the concepts of Hausdorff measure and Hausdorff dimension. Therefore the Hausdorff dimension is sometimes called Hausdorff-Besicovitch dimension.

Clearly the Hausdorff dimension is not necessarily an integer. Perhaps the most famous example of a set with non-integer ${\rm dim}_H$ is the Cantor set $C$, for which we have ${\rm dim}_H (C) = (\ln 2)/(\ln 3)$ (cp. with Section 4.10 of [Ma]). The construction in Section 4.13 of [Ma] leads easily to subsets of the euclidean space with arbitrary Hausdorff dimension.

If $(X,d)$ is a metric space and $Y\subset X$, we can then restrict the metric $d$ on $Y\times Y$, consider the resulting metric space and define the Hausdorff dimension of any $E\subset Y$ as a subset of $Y$. It is easy to see that this does not change the result: i.e. the Hausdorff dimension of $E$ as a subset of $Y$ or as a subset of $X$ is the same.

### Properties[edit]

* If $\psi: X\to Y$ is a Lipschitz map, then the Hausdorff dimension of $\psi (A)$ is at most that of $A$.
* If $A$ is a countable union of sets $A_i$'s, the Hausdorff dimension of $A$ is the supremum of the Hausdorff dimensions of the $A_i$'s.
* The Hausdorff dimension of $A\times B$ is at least the sum of the Hausdorff dimensions of the spaces $A$ and $B$ and it is not necessarily equal to the sum.
* The Hausdorff dimension of a Riemannian manifold corresponds to its topological dimension.

For all these facts we refer to [Ma]. A useful tool to estimate the Hausdorff dimension of Borel subsets of the euclidean space is Frostman's Lemma.

### Other definitions of dimension[edit]

For general metric spaces one can define the metric dimension (see [HW]), whereas for subsets of the euclidean space one can define the Minkowski dimension and the packing dimension (see [Ma]). For general sets these dimensions do not coincide.

### References[edit]

[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)
[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