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
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
Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area.
In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer.
They appear naturally in harmonic analysis or potential theory.
Define where the infimum is over all countable covers of
is, the more collections of sets are permitted, making the infimum not larger.
By Carathéodory's extension theorem, its restriction to the σ-field of Carathéodory-measurable sets is a measure.
Due to the metric outer measure property, all Borel subsets of
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
restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.
Note that if d is a positive integer, the d-dimensional Hausdorff measure of
is a rescaling of the usual d-dimensional Lebesgue measure
, which is normalized so that the Lebesgue measure of the unit cube [0,1]d is 1.
In fact, for any Borel set E, where αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function This is where
is the volume of the unit diameter d-ball.
Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that the value
, so that Hausdorff d-dimensional measure coincides exactly with Lebesgue measure in the case of Euclidean space.
That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to the idea that the area of a line is zero and the length of a 2D shape is in some sense infinity.
This leads to one of several possible equivalent definitions of the Hausdorff dimension: where we take
Note that it is not guaranteed that the Hausdorff measure must be finite and nonzero for some d, and indeed the measure at the Hausdorff dimension may still be zero; in this case, the Hausdorff dimension still acts as a change point between measures of zero and infinity.
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.
-dimensional Minkowski content of a closed
-dimensional Hausdorff measure (Federer 1969, Theorem 3.2.29).
For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdorff measure is zero.
Examples of gauge functions include The former gives almost surely positive and