User:OdedSchramm/hm

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In mathematics a Hausdorff measure assigns a number in [0,\infty] to every metric space. The zero dimensional Hausdorff measure of a metric space is the number of points in the space (if the space is finite) or \infty if the space is infinite. The one dimensional Hausdorff measure of a metric space which is an imbedded path in \R^n is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d\ge 0 which is not necessarily an integer. These measures are useful for studying the size of fractals.

[edit] Definition

Fix some d\ge 0 and a metric space X. Let S\subset X be any subset of X. For δ > 0 let

H^d_\delta(S):=\inf\Bigl\{\sum_i r_i^d : \text{there is a collection of balls with radii }r_i\in(0,\delta)\text{ which cover }S\Bigr\}.

Note that H^d_\delta(S) is monotone decreasing in δ since the larger δ is, the more collections of balls are permitted. Thus, the limit \lim_{\delta\to 0}H^d_\delta(S) exists. Set

H^d(S):=\sup_{\delta>0} H^d_\delta(S)=\lim_{\delta\to 0}H^d_\delta(S).

This is the d-dimensional Hausdorff measure of S.

[edit] Properties of Hausdorff measures

The Hausdorff measures Hd are outer measures. Moreover, all Borel subsets of X are Hd measureable. In particular, the theory of outer measures implies that Hd is countably additive on the Borel σ-field.

[edit] References

  • L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992
  • H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
  • Frank Morgan, Geometric Measure Theory, Academic Press, 1988. Good introductory presentation with lots of illustrations.
  • E. Szpilrajn, La dimension et la mesure, Fundamenta Mathematica 28, 1937, pp 81-89.