# Mean-shift

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

## Definition

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

and

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

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

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 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

## References

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

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