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.
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Definition
Let
Let
(The infimum is over all countable covers of
Note that
It can be seen that
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
Properties of Hausdorff measures
Note that if d is a positive integer, the d dimensional Hausdorff measure of Rd is a rescaling of usual d-dimensional Lebesgue measure
where αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function
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
where we take
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
In fractal geometry, some fractals with Hausdorff dimension
This is the Hausdorff measure of