The universal Glivenko-Cantelli property

• 成果类型: Article
• 署名作者: van Handel, Ramon
• 署名单位: Princeton University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0416-5
• 发表日期: 2013
• 页码: 911-934
• 关键词: uniform-convergence; reals

摘要

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space (X, X, and let N-[](F, epsilon, mu) be the smallest number of epsilon-brackets in L-1(mu) needed to cover F. The following are equivalent: F is a universal Glivenko-Cantelli class. N-[](F, epsilon, mu) < infinity for every epsilon > 0 and every probability measure mu. F is totally bounded in L-1(mu) for every probability measure mu. F does not contain a Boolean sigma-independent sequence. It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.