Recouvrements, derivation des mesures et dimensions

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. splitting into m disjoint families). Since Besicovitch, covering properties are known to be the main tool for providing derivation theorems for any pair of measures on (X,d).Assuming that any ball for d belongs to the Baire s-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 Nagata-dimension (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 Rn with any given norm (n > 1), they are > n. For spaces with nonnegative curvature = 0 (for example for Rn 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 Rn) and try to obtain the exact degree in R and R2