Hausdorff density

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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


Let be a Radon measure and some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,


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[edit]

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

Marstrand's theorem: Let be a Radon measure on . Suppose that the s-density exists and is positive and finite for a in a set of positive measure. Then s is an integer.

Preiss' theorem[edit]

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.

Preiss' theorem: Let be a Radon measure on . Suppose that m is an integer and the m-density exists and is positive and finite for almost every a in the support of . Then is m-rectifiable, i.e. ( is absolutely continuous with respect to Hausdorff measure ) and the support of is an m-rectifiable set.

External links[edit]