Hausdorff measure

It is related to, although different from, the Hausdorff measure.

He also wrote some papers on the Hausdorff measure during the War.

This is the construction used in the definition of Hausdorff measures for a metric space.

Thus, the concept of the Hausdorff measure generalizes counting, length, and area.

In fact, there are d-dimensional Hausdorff measures for any d 0, which is not necessarily an integer.

It is called the -dimensional Hausdorff measure of .

The n-dimensional Hausdorff measure yields a density, as above.

The Hausdorff measure by comparison, is countable additive.

H is the n-dimensional Hausdorff measure.

The Hausdorff measure is not to be confused with the notion of Hausdorff dimension.

This is the Hausdorff measure of with gauge function , or -Hausdorff measure.

In potential theory, a result that estimates the Hausdorff measure of the set on which a logarithmic Newtonian potential is small.

Sometimes the Hausdorff dimension and Hausdorff measure are used to "measure" the size of such curves.

The one dimensional Hausdorff measure of a simple curve in R is equal to the length of the curve.

He introduced the concepts now called Hausdorff measure and Hausdorff dimension, which have been useful in the theory of fractals.

Hausdorff Measures, Cambridge University Press, 1998.

In order to "measure" the "size" of such sets, mathematicians have considered the following variation on the notion of the Hausdorff measure:

Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of s-dimensional Hausdorff measure.

A function times this Hausdorff measure can then be integrated over k-dimensional subsets, providing a measure-theoretic analog to integration of k-forms.

(Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).

Let dim denote Hausdorff dimension and H denote 1-dimensional Hausdorff measure.

Likewise, the two dimensional Hausdorff measure of a measurable subset of R is proportional to the area of the set.

In fractal geometry, some fractals with Hausdorff dimension have zero or infinite -dimensional Hausdorff measure.

Then is m-rectifiable, i.e. ( is absolutely continuous with respect to Hausdorff measure ) and the support of is an m-rectifiable set.

A Borel set is rectifiable if , where denotes one-dimensional Hausdorff measure restricted to the set .