Hausdorff density explained

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

Definition

Let

\mu be a Radon measure and a\in R ^{n} some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively, \Theta ^{*s} (\mu,a) = \limsup _{r →} (\mu(B _{r} (a)))/(r ^{s}) and\Theta ^{s} _{*} (\mu,a) = \liminf _{r →} (\mu(B _{r} (a)))/(r ^{s}) where B _{r} (a) is the ball of radius r > 0 centered at a. Clearly, \Theta ^{s} _{*} (\mu,a)\leq \Theta ^{*s} (\mu,a) for all a\in R ^{n} . In the event that the two are equal, we call their common value the s-density of \mu at a and denote it \Theta ^{s} (\mu,a) .

Marstrand's theorem

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

Marstrand's theorem: Let

\mu be a Radon measure on R ^{d} . Suppose that the s-density \Theta ^{s} (\mu,a) exists and is positive and finite for a in a set of positive \mu measure. Then s is an integer.

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.

Preiss' theorem: Let

\mu be a Radon measure on R ^{d} . Suppose that m\geq 1 is an integer and the m-density \Theta ^{m} (\mu,a) exists and is positive and finite for \mu almost every a in the support of \mu . Then \mu is m-rectifiable, i.e. \mu\ll H ^{m} (\mu is absolutely continuous with respect to Hausdorff measure H ^{m} ) and the support of \mu is an m-rectifiable set.

External links

• Density of a set at Encyclopedia of Mathematics
• Rectifiable set at Encyclopedia of Mathematics

References

• Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.
• Preiss . David . David Preiss . Geometry of measures in R ^{n} : distribution, rectifiability, and densities . 1971410 . Ann. Math. . 125 . 3 . 537 - 643 . 1987 . 10.2307/1971410. 10338.dmlcz/133417 . free.