View source for Hausdorff dimension

{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

A numerical invariant of a metric spaces, introduced by F. Hausdorff in {{Cite|Ha}}. 

===Defininition===
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) := 
\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 measure|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 measure|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 $\mathcal{H}$ is an [[Outer measure|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 {{Cite|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===
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 ${\rm dim}_H (C) = (\ln 3)/(\ln 2)$ (cp. with Section 4.10 of {{Cite|Ma}}). The construction in Section 4.13 lead 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===
* 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 {{Cite|Ma}}. A useful tool to estimate the Hausdorff dimension of [[Borel set|Borel subsets]] of the euclidean space is Frostman's Lemma
(see [[Hausdorff measure]]).

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

===References===
{|
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|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|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}}
|-
|}