Recouvrements, derivation des mesures et dimensions
Recouvrements, derivation des mesures et dimensions
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回收、测量和尺寸的推导
DOI：
10.4171/rmi/478
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发表时间：
2006
影响因子：
1.2
通讯作者：
T. Q. D. Gromard
中科院分区：
文献类型：
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作者：
P. Assouad;T. Q. D. Gromard
设X是具有对称核d(不一定是距离)的集合。称空间(X，d)具有弱(分别为。强)次=m的覆盖性质[简记为PRF(M)(分别Prf(M))]，如果对于(X，d)的每个闭球的族B，半径按递减序列(分别为有界半径)，有一个子族，覆盖B的每个元素的中心，且阶为m(分别为.分裂成m个不相交的家族)。自Besicovitch以来，覆盖性质被认为是为(X，d)上的任何一对测度提供导数定理的主要工具。
假设d的任一球都属于d的Baire S-代数，我们证明了PRF蕴含着一个几乎必然的导数定理。当(X，d)是完全可分度量空间时，D.Preiss提出了这一蕴涵。在更强的可测性假设下(将在本文后面说明)，我们证明了限制于常半径球的PRF蕴含着一个关于测度收敛的导数定理。
我们很容易地证明了一个等价于PRF(m+1)(分别.对于仅限于具有恒定半径的球的PRF(m+1)而言)是Nagata维(分别.(X，d)的de Groot维度)=m。这两个维度(见J.I.Nagata)不小于拓扑维；对于具有任何给定范数(n>；1)的Rn，它们是>；n。对于具有非负曲率=0的空间(例如，对于具有任何给定范数的Rn)，我们将这些维度表示为网的基数；在这些空间中，我们给出了PRF的次数的一个类似的上界(推广了Furedi和Loeb关于Rn的一个结果)，并试图得到R和R2中的精确次数
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 s-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