Recouvrements, derivation des mesures et dimensions

  • Patrice Assouad

    Université Paris-Sud 11, Orsay, France
  • Thierry Quentin de Gromard

    Université Paris-Sud 11, Orsay, France


Let be a set with a symmetric kernel (not necessarily a distance). The space is said to have the weak (resp. strong) covering property of degree [briefly \textbf{prf} (resp. \textbf{prF})], if, for each family of closed balls of with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of , and of order (resp. spliting into disjoint families). Since Besicovitch, covering properties are known to be the main tool for proving derivation theorems for any pair of measures on . Assuming that any ball for belongs to the Baire -algebra for , we show that the \textbf{prf} implies an almost sure derivation theorem. This implication was stated by D. Preiss when is a complete separable metric space. With stronger measurability hypothesis (to be stated later in this paper), we show that the \textbf{prf} restricted to balls with constant radius implies a derivation theorem with convergence in measure. We show easily that an equivalent to the \textbf{prf} (resp. to the \textbf{prf} restricted to balls with constant radius) is that the Nagata-dimension (resp. the De Groot-dimension) of is . These two dimensions (see J.I. Nagata) are not lesser than the topological dimension; for with any given norm (), they are . For spaces with nonnegative curvature (for example for 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 \textbf{prF} (generalizing a result of Furedi and Loeb for ) and try to obtain the exact degree in and .

Cite this article

Patrice Assouad, Thierry Quentin de Gromard, Recouvrements, derivation des mesures et dimensions. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 893–953

DOI 10.4171/RMI/478