Recouvrements, derivation des mesures et dimensions
Revista matemática iberoamericana
Let X be a set with a symmetric kernel d (not necessarily a distance). The space (X, d) is said to have the weak (resp. strong) covering property of degree ≤ m [briefly prf (m) (resp. prF(m))], if, for each family B of closed balls of (X, d) with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of B , and of order ≤ m (resp. spliting into m disjoint families). Since Besicovitch, covering properties are known to be the main tool
... to be the main tool for proving derivation theorems for any pair of measures on (X, d). Assuming that any ball for d belongs to the Baire σ-algebra for d, we show that the prf implies an almost sure derivation theorem. This implication was stated by D. Preiss when (X, d) is a complete separable metric space. With stronger measurability hypothesis (to be stated later in this paper), we show that the prf restricted to balls with constant radius implies a derivation theorem with convergence in measure. We show easily that an equivalent to the prf (m + 1) (resp. to the prf (m+1) restricted to balls with constant radius) is that the Nagatadimension (resp. the De Groot-dimension) of (X, d) is ≤ m. These two dimensions (see J.I. Nagata) are not lesser than the topological dimension ; for R n with any given norm (n > 1), they are > n. For spaces with nonnegative curvature ≥ 0 (for example for R n with any given norm), we express these dimensions as the cardinality of a net ; in these spaces, we give a similar upper bound for the degree of the prF (generalizing a result of Furedi and Loeb for R n ) and try to obtain the exact degree in R and R 2 . 2000 Mathematics Subject Classification: 28A15.