Hausdorff density

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

Definition

Let ${\displaystyle \mu }$ be a Radon measure and ${\displaystyle a\in \mathbb {R} ^{n}}$ some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,

${\displaystyle \Theta ^{*s}(\mu ,a)=\limsup _{r\rightarrow 0}{\frac {\mu (B_{r}(a))}{r^{s}}}}$

and

${\displaystyle \Theta _{*}^{s}(\mu ,a)=\liminf _{r\rightarrow 0}{\frac {\mu (B_{r}(a))}{r^{s}}}}$

where ${\displaystyle B_{r}(a)}$ is the ball of radius r > 0 centered at a. Clearly, ${\displaystyle \Theta _{*}^{s}(\mu ,a)\leq \Theta ^{*s}(\mu ,a)}$ for all ${\displaystyle a\in \mathbb {R} ^{n}}$. In the event that the two are equal, we call their common value the s-density of ${\displaystyle \mu }$ at a and denote it ${\displaystyle \Theta ^{s}(\mu ,a)}$.

Marstrand's theorem

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

Marstrand's theorem: Let ${\displaystyle \mu }$ be a Radon measure on ${\displaystyle \mathbb {R} ^{d}}$. Suppose that the s-density ${\displaystyle \Theta ^{s}(\mu ,a)}$ exists and is positive and finite for a in a set of positive ${\displaystyle \mu }$ 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 ${\displaystyle \mu }$ be a Radon measure on ${\displaystyle \mathbb {R} ^{d}}$. Suppose that m${\displaystyle \geq 1}$ is an integer and the m-density ${\displaystyle \Theta ^{m}(\mu ,a)}$ exists and is positive and finite for ${\displaystyle \mu }$ almost every a in the support of ${\displaystyle \mu }$. Then ${\displaystyle \mu }$ is m-rectifiable, i.e. ${\displaystyle \mu \ll H^{m}}$ (${\displaystyle \mu }$ is absolutely continuous with respect to Hausdorff measure ${\displaystyle H^{m}}$) and the support of ${\displaystyle \mu }$ 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 (1987). "Geometry of measures in ${\displaystyle R^{n}}$: distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410. hdl:10338.dmlcz/133417. JSTOR 1971410.