Hausdorff Measure - Generalizations

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 is said to be -rectifiable if it is the image of a bounded set in under a Lipschitz function. If, then the -dimensional Minkowski content of a closed -rectifiable subset of is equal to times the -dimensional Hausdorff measure (Federer 1969, Theorem 3.2.29).

In fractal geometry, some fractals with Hausdorff dimension have zero or infinite -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdoff measure is zero. In order to “measure” the “size” of such sets, mathematicians have considered the following variation on the notion of the Hausdorff measure:

In the definition of the measure is replaced with, where is any monotone increasing set function satisfying .

This is the Hausdorff measure of with gauge function, or -Hausdorff measure. A -dimensional set may satisfy, but with an appropriate Examples of gauge functions include or . The former gives almost surely positive and -finite measure to the Brownian path in when, and the latter when .