2010 Mathematics Subject Classification: Primary: 28A Secondary: 37F35 [MSN][ZBL]
A numerical invariant of a metric space, introduced by F. Hausdorff in [Ha].
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
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*}
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.
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.
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.
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