Open Access
December, 2006 Recouvrements, derivation des mesures et dimensions
Patrice Assouad , Thierry Quentin de Gromard
Rev. Mat. Iberoamericana 22(3): 893-953 (December, 2006).


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 $\leq m$ [briefly \textbf{prf}$(m)$ (resp. \textbf{prF}$(m)$)], if, for each family $\mathcal{B}$ 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 $\mathcal{B}$, and of order $\leq m$ (resp. spliting into $m$ disjoint families). Since Besicovitch, covering properties are known to be the main tool for proving derivation theorems for any pair of measures on $(X,d)$. Assuming that any ball for $d$ belongs to the Baire $\sigma$-algebra for $d$, we show that the \textbf{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 \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}$(m+1)$ (resp. to the \textbf{prf}$(m+1)$ restricted to balls with constant radius) is that the Nagata-dimension (resp. the De Groot-dimension) of $(X,d)$ is $\leq m$. These two dimensions (see J.I. Nagata) are not lesser than the topological dimension; for $\mathbb{R}^n$ with any given norm ($n > 1$), they are $> n$. For spaces with nonnegative curvature $\geq 0$ (for example for $\mathbb{R}^n$ 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 $\mathbb{R}^n$) and try to obtain the exact degree in $\mathbb{R}$ and $\mathbb{R}^2$.


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Patrice Assouad . Thierry Quentin de Gromard . "Recouvrements, derivation des mesures et dimensions." Rev. Mat. Iberoamericana 22 (3) 893 - 953, December, 2006.


Published: December, 2006
First available in Project Euclid: 22 January 2007

zbMATH: 1123.28001
MathSciNet: MR2320406

Primary: ‎28A15

Keywords: covering properties , derivation theorems , dimensions of metric spaces

Rights: Copyright © 2006 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.22 • No. 3 • December, 2006
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