# Mean-shift

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

and

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

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

## Preiss' theorem

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

**Preiss' theorem:**Let $\mu$ be a Radon measure on $\mathbb{\mathbb{R}}}^{d$. Suppose that*m*$\ge 1$ is an integer and the*m*-density ${\mathrm{\Theta}}^{m}(\mu ,a)$ 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

## References

- Pertti Mattila,
*Geometry of sets and measures in Euclidean spaces.*Cambridge Press, 1995.

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