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Covering, measure derivation and dimensions. (Recouvrements, derivation des mesures et dimensions.) (French. English summary) Zbl 1123.28001

The paper generalizes classical work of Besicovitch about derivation of measures by suitable coverings from Euclidean spaces to a space \(X\) with symmetric kernel \(d\), i.e. \(d\) has all properties of a semi-metric with the exception of the triangle inequality. Besides others, weak or strong covering properties of degree less than or equal to \(m\) are the most basic ones to be studied in this paper. These coverings are defined by the property that each family of closed balls with radii in a decreasing sequence or with bounded radii contains a finite subfamily covering the center of each element at most \(m\) times, respectively. Both coverings are equivalent if the symmetric kernel has the so-called ‘doubling property’ and if its balls are sufficiently regular.
If all balls belong to the Baire field of the basic symmetric kernel, an almost everywhere derivation theorem holds true. This generalizes classical work of M. de Guzman for Euclidean spaces as well as of D. Preiss for weak coverings of complete, separable metric spaces. As a novelty, a theorem on the derivation in measure is obtained under stronger measurabilty properties for weak coverings restricted to balls with constant radii. The proof of this result uses ideas of Mattila.
Like D. Preiss the authors study relations between coverings and dimension. It is easily seen, that the weak coverings of degree less than or equal to \(m+ 1\) is equivalent to the fact, that \((X,d)\) has Nagata dimension less than or equal to \(m\), and if the balls are of constant radii, that its De Groot dimension is less than or equal to \(m\). As an application the finiteness of the De Groot dimension for the Heisenberg group under its usual metric follows, its Nagata dimension being infinite. Another application is for the \(n\)-dimensional Euclidean space, constructing on it a symmetric kernel which is equivalent to the Euclidean distance, being the minimum of \(n+1\) ultrametrics, and having open balls being open for the Euclidean topology.
Finally, the authors anounce generalizations for open balls and ‘pseudoballs’ to appear elsewhere.

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
54E99 Topological spaces with richer structures
54F45 Dimension theory in general topology
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