JournalsrmiVol. 22, No. 3pp. 893–953

Recouvrements, derivation des mesures et dimensions

  • Patrice Assouad

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

    Université Paris-Sud 11, Orsay, France
Recouvrements, derivation des mesures et dimensions cover
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Abstract

Let XX be a set with a symmetric kernel dd (not necessarily a distance). The space (X,d)(X,d) is said to have the weak (resp. strong) covering property of degree m\leq m [briefly \textbf{prf}(m)(m) (resp. \textbf{prF}(m)(m))], if, for each family B\mathcal{B} of closed balls of (X,d)(X,d) with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of B\mathcal{B}, and of order m\leq m (resp. spliting into mm disjoint families). Since Besicovitch, covering properties are known to be the main tool for proving derivation theorems for any pair of measures on (X,d)(X,d). Assuming that any ball for dd belongs to the Baire σ\sigma-algebra for dd, we show that the \textbf{prf} implies an almost sure derivation theorem. This implication was stated by D. Preiss when (X,d)(X,d) 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}(m+1)(m+1) (resp. to the \textbf{prf}(m+1)(m+1) restricted to balls with constant radius) is that the Nagata-dimension (resp. the De Groot-dimension) of (X,d)(X,d) is m\leq m. These two dimensions (see J.I. Nagata) are not lesser than the topological dimension; for Rn\mathbb{R}^n with any given norm (n>1n > 1), they are >n> n. For spaces with nonnegative curvature 0\geq 0 (for example for Rn\mathbb{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 \textbf{prF} (generalizing a result of Furedi and Loeb for Rn\mathbb{R}^n) and try to obtain the exact degree in R\mathbb{R} and R2\mathbb{R}^2.

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