Page "Hausdorff measure" Paragraph 0

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R < sup > n </ sup > 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 R < sup > n </ sup > is equal to the length of the curve.

Likewise, the two dimensional Hausdorff measure of a measurable subset of R < sup > 2 </ sup > 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.