Coverings, derivation of measures and dimensions.

Patrice Assouad; Thierry Quentin de Gromard

Revista Matemática Iberoamericana (2006)

  • Volume: 22, Issue: 3, page 893-953
  • ISSN: 0213-2230


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 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 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.

How to cite


Assouad, Patrice, and Quentin de Gromard, Thierry. "Recouvrements, derivation des mesures et dimensions.." Revista Matemática Iberoamericana 22.3 (2006): 893-953. <>.

author = {Assouad, Patrice, Quentin de Gromard, Thierry},
journal = {Revista Matemática Iberoamericana},
keywords = {Teoría de la medida; Recubrimiento; Espacios métricos; derivation of measures; symmetric kernel; coverings; Nagata dimension; De Groot dimension},
language = {fre},
number = {3},
pages = {893-953},
title = {Recouvrements, derivation des mesures et dimensions.},
url = {},
volume = {22},
year = {2006},

AU - Assouad, Patrice
AU - Quentin de Gromard, Thierry
TI - Recouvrements, derivation des mesures et dimensions.
JO - Revista Matemática Iberoamericana
PY - 2006
VL - 22
IS - 3
SP - 893
EP - 953
LA - fre
KW - Teoría de la medida; Recubrimiento; Espacios métricos; derivation of measures; symmetric kernel; coverings; Nagata dimension; De Groot dimension
UR -
ER -

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