Hausdorffmaat

Hausdorff measure is a concept in fractal geometry and measure theory. It is a way to assign a "size" or "measure" to sets of points in Euclidean space, particularly useful for sets that are not easily measured by traditional methods like length, area, or volume. The Hausdorff measure is parameterized by a dimension, denoted by $s$. For a given set $E$ and a dimension $s$, the $s$-dimensional Hausdorff measure of $E$, written as $H^s(E)$, quantifies the set's "size" in $s$ dimensions.

The definition of Hausdorff measure involves covering the set $E$ with a collection of small sets, each with a diameter no larger than a given value $\delta$. For each such cover, one sums the $s$-th power of the diameters of the covering sets. The $s$-dimensional Hausdorff measure is then defined as the infimum of these sums over all possible $\delta$-covers, as $\delta$ approaches zero. This process allows for the measurement of sets with fractional dimensions, which are characteristic of fractals.

A key property of the Hausdorff measure is its behavior with respect to the dimension. For a given set $E$, there exists a critical dimension, called the Hausdorff dimension of $E$, denoted by $\dim_H(E)$. If $s < \dim_H(E)$, then $H^s(E) = \infty$. If $s > \dim_H(E)$, then $H^s(E) = 0$. At the Hausdorff dimension itself, the measure $H^{\dim_H(E)}(E)$ can be a finite positive number, zero, or infinity, depending on the specific set. This phenomenon highlights how Hausdorff measure can distinguish between sets with similar topological dimensions but different fractal structures.