Hausdorff Density

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 $a\in\mathbb^$ some point in Euclidean space. The ''s''-dimensional upper and lower Hausdorff densities are defined to be, respectively,
:$\Theta^(\mu,a)=\limsup_\frac$
and
:$\Theta_^(\mu,a)=\liminf_\frac$
where $B_(a)$ is the ball of radius ''r'' > 0 centered at ''a''. Clearly, $\Theta_^(\mu,a)\leq \Theta^(\mu,a)$ for all $a\in\mathbb^$. In the event that the two are equal, we call their common value the s-density of $\mu$ at ''a'' and denote it $\Theta^(\mu,a)$.

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 $\mathbb^$. Suppose that the ''s''-density $\Theta^(\mu,a)$ 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 $\mathbb^$. Suppose that ''m''$\geq 1$ is an integer and the ''m''-density $\Theta^(\mu,a)$ exists and is positive and finite for $\mu$ almost every ''a'' in the support of $\mu$. Then $\mu$ is ''m''-rectifiable, i.e. $\mu\ll H^$ ($\mu$ 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
a
Encyclopedia of Mathematics

Rectifiable set
a
Encyclopedia of Mathematics

References

* Pertti Mattila, ''Geometry of sets and measures in Euclidean spaces.'' Cambridge Press, 1995.
* {{cite journal
, last = Preiss , first = David
, author-link = David Preiss
, title = Geometry of measures in $R^n$: distribution, rectifiability, and densities
, jstor = 1971410
, journal = Ann. Math. , volume = 125
, issue = 3 , pages = 537–643 , year = 1987 , doi=10.2307/1971410
, hdl = 10338.dmlcz/133417
, hdl-access = free

Measure theory