Hausdorff measure

In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in 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. Likewise, the one-dimensional Hausdorff measure of a simple curve in is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of 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. Let be a metric space. For any subset , let denote its diameter, that is Let be any subset of and a real number. Define where the infimum is over all countable covers of by sets satisfying . Note that is monotone nonincreasing in since the larger is, the more collections of sets are permitted, making the infimum not larger. Thus, exists but may be infinite. Let It can be seen that is an outer measure (more precisely, it is a metric outer measure). By Carathéodory's extension theorem, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the -dimensional Hausdorff measure of . Due to the metric outer measure property, all Borel subsets of are measurable. In the above definition the sets in the covering are arbitrary. However, we can require the covering sets to be open or closed, or in normed spaces even convex, that will yield the same numbers, hence the same measure. In restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.