Recouvrements, derivation des mesures et dimensions

Abstract

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 $\le m$ [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 $\le m$ (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 $\le m$. 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 $\ge 0$ (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 ${\mathbb{R}}^2$.