Hausdorff density


In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.

Definition

Let be a Radon measure and some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
and
where is the ball of radius r > 0 centered at a. Clearly, for all. In the event that the two are equal, we call their common value the s-density of at a and denote it.

Marstrand's theorem

The following theorem states that the times when the s-density exists are rather seldom.

Preiss' theorem

In 1987 David Preiss proved a stronger version of Marstrand's theorem. One consequence is that sets with positive and finite density are rectifiable sets.