Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rⁿ or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. The one-dimensional Hausdorff measure of a simple curve in Rⁿ is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R² is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory.

Definition

Let $(X,\rho )$ be a metric space. For any subset $\scriptstyle U\subset X$ , let ${\mathrm {diam}}\;U$ denote its diameter, that is

{\mathrm {diam}}\;U:=\sup\{\rho (x,y)|x,y\in U\},\quad {\mathrm {diam}}\;\emptyset :=0

Let $S$ be any subset of $X$ , and $\delta >0$ a real number. Define

H_{\delta }^{d}(S)=\inf {\Bigl \{}\sum _{{i=1}}^{\infty }(\operatorname {diam}\;U_{i})^{d}:\bigcup _{{i=1}}^{\infty }U_{i}\supseteq S,\,\operatorname {diam}\;U_{i}<\delta {\Bigr \}}.

(The infimum is over all countable covers of $S$ by sets $\scriptstyle U_{i}\subset X$ satisfying $\scriptstyle \operatorname {diam}\;U_{i}<\delta$ .)

Note that $\scriptstyle H_{\delta }^{d}(S)$ is monotone decreasing in $\delta$ since the larger $\delta$ is, the more collections of sets are permitted, making the infimum smaller. Thus, the limit $\scriptstyle \lim _{{\delta \to 0}}H_{\delta }^{d}(S)$ exists but may be infinite. Let

H^{d}(S):=\sup _{{\delta >0}}H_{\delta }^{d}(S)=\lim _{{\delta \to 0}}H_{\delta }^{d}(S).

It can be seen that $H^{d}(S)$ is an outer measure (more precisely, it is a metric outer measure). By general theory, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the $d$ -dimensional Hausdorff measure of $S$ . Due to the metric outer measure property, all Borel subsets of $X$ are $H^{d}$ measurable.

In the above definition the sets in the covering are arbitrary. However, they may be taken to be open or closed, and will yield the same measure, although the approximations $\scriptstyle H_{\delta }^{d}(S)$ may be different (Federer 1969, §2.10.2). If $X$ is a normed space the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different measure.^[1]

Properties of Hausdorff measures

Note that if d is a positive integer, the d dimensional Hausdorff measure of R^d is a rescaling of usual d-dimensional Lebesgue measure $\lambda _{d}$ which is normalized so that the Lebesgue measure of the unit cube [0,1]^d is 1. In fact, for any Borel set E,

\lambda _{d}(E)=2^{{-d}}\alpha _{d}H^{d}(E)\,

where α_d is the volume of the unit d-ball; it can be expressed using Euler's gamma function

\alpha _{d}={\frac {\Gamma ({\frac 12})^{d}}{\Gamma ({\frac {d}{2}}+1)}}={\frac {\pi ^{{d/2}}}{\Gamma ({\frac {d}{2}}+1)}}.

Remark. Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that it is normalized in such a way that Hausdorff d-dimensional measure in the case of Euclidean space coincides exactly with Lebesgue measure.

Relation with Hausdorff dimension

One of several possible equivalent definitions of the Hausdorff dimension is

\operatorname {dim}_{{{\mathrm {Haus}}}}(S)=\inf\{d\geq 0:H^{d}(S)=0\}=\sup {\bigl (}\{d\geq 0:H^{d}(S)=\infty \}\cup \{0\}{\bigr )},

where we take

\inf \emptyset =\infty .\,

Generalizations

In geometric measure theory and related fields, the Minkowski content is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of $\scriptstyle\mathbb{R}^n$ is said to be $m$ -rectifiable if it is the image of a bounded set in $\scriptstyle\mathbb{R}^n$ under a Lipschitz function. If $m<n$ , then the $m$ -dimensional Minkowski content of a closed $m$ -rectifiable subset of $\scriptstyle\mathbb{R}^n$ is equal to $2^{{-m}}\alpha _{m}$ times the $m$ -dimensional Hausdorff measure (Federer 1969, Theorem 3.2.29).

In fractal geometry, some fractals with Hausdorff dimension $d$ have zero or infinite $d$ -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdoff measure is zero. In order to “measure” the “size” of such sets, mathematicians have considered the following variation on the notion of the Hausdorff measure:

In the definition of the measure

|U_{i}|^{d}

is replaced with

\phi (U_{i})

, where

\phi

is any monotone increasing set function satisfying

\phi (\emptyset )=0

This is the Hausdorff measure of $S$ with gauge function $\phi$ , or $\phi$ -Hausdorff measure. A $d$ -dimensional set $S$ may satisfy $H^{d}(S)=0$ , but $\scriptstyle H^{\phi }(S)\in (0,\infty )$ with an appropriate $\phi .$ Examples of gauge functions include $\scriptstyle \phi (t)=t^{2}\,\log \log {\frac 1t}$ or $\scriptstyle \phi (t)=t^{2}\log {\frac {1}{t}}\log \log \log {\frac {1}{t}}$ . The former gives almost surely positive and $\sigma$ -finite measure to the Brownian path in $\scriptstyle\mathbb{R}^n$ when $n>2$ , and the latter when $n=2$ .

External links

References

↑ Yeh, J. (2006), Real analysis: theory of measure and integration, p. 681

Evans, Lawrence C.; Gariepy, Ronald F. (1992), Measure Theory and Fine Properties of Functions, CRC Press .
Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4 .
Yan Kun (2007), Fractal Measure.
Hausdorff, Felix (1918), "Dimension und äusseres Mass", Mathematische Annalen, 79 (1-2): 157–179, doi:10.1007/BF01457179 .
Morgan, Frank (1988), Geometric Measure Theory, Academic Press .
Szpilrajn, E (1937), "La dimension et la mesure" (PDF), Fundamenta Mathematicae, 28: 81–89 .

This article is issued from Wikipedia - version of the 10/20/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.