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 edit
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 edit
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 edit
- 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.