Recouvrements, derivation des mesures et dimensions
Patrice Assouad2000, Revista Matemática Iberoamericana
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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 ≤ 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. 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 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 Nagatadimension (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 R n with any given norm (n > 1), they are > n. For spaces with nonnegative curvature ≥ 0 (for example for 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 prF (generalizing a result of Furedi and Loeb for R n) and try to obtain the exact degree in R and R 2 .
