Hausdorffmértékek

Hausdorffmértékek are a fundamental concept in geometric measure theory, providing a way to assign a "size" or "measure" to sets of points in Euclidean space. These measures are particularly useful for dealing with irregular or fractal shapes, where traditional notions of length, area, or volume may not apply.

The core idea of a Hausdorffmérték is to cover a set with small balls and consider the infimum of the sum of their diameters raised to a certain power. This power, denoted by 's', is crucial. For a given set E and a non-negative real number s, the s-dimensional Hausdorffmérték, denoted by H^s(E), quantifies the size of E in a way that depends on 's'.

If s is an integer, H^s(E) often coincides with the familiar Lebesgue measure (length for s=1, area for s=2, volume for s=3) for "regular" sets. However, the power of Hausdorffmértékek lies in their ability to handle non-integer dimensions. For fractal sets, the Hausdorff dimension is the value of 's' at which the Hausdorffmérték transitions from infinity to zero. This dimension often provides a more accurate description of the set's complexity than its topological dimension.

The existence and properties of Hausdorffmértékek are established through a rigorous mathematical framework. They are countably subadditive and, under certain conditions, can be used to define distances and study the geometry of complex sets.