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In mathematics a Hausdorff measure assigns a number in 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 if the space is infinite. The one dimensional Hausdorff measure of a metric space which is an imbedded path in is proportional to the length of the path. Likewise, the two dimensional Hausdorff measure of a subset of 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 -dimensional Hausdorff measures for any which is not necessarily an integer. These measures are useful for studying the size of fractals.

Definition

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Fix some and a metric space . Let be any subset of . For let

Note that is monotone decreasing in since the larger is, the more collections of balls are permitted. Thus, the limit exists. Set

This is the -dimensional Hausdorff measure of .

Properties of Hausdorff measures

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

References

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  • 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.