Relative Compactness for Capacities, Measures, Upper Semicontinuous Functions and Closed Sets

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• 作者：George L. O'Brien and Stephen Watson
• 关键词：Capacities ; narrow topology ; relative compactness ; equitightness ; Prohorov''s theorem ; Alexander''s subbase theorem
• 刊名：Journal of Theoretical Probability
• 出版年：1998
• 出版时间：1998
• 年：1998
• 卷：11
• 期：3
• 页码：577-588
• 全文大小：589 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

We prove an Ascoli theorem for capacities. This theorem which characterizes relatively compact sets of capacities is widely applicable and many Ascoli theorems for particular classes of capacities can immediately be deduced as corollaries. Indeed it is usually necessary only to demonstrate that these classes are closed and then to simplify the characterization when possible. In particular, we show that the proof of the classical Prohorov theorem can be naturally factored into the shorter proof of the Ascoli theorem for capacities and into the somewhat longer proof that the class of probability measures is closed in the class of capacities. We also deduce new and known Ascoli theorems for sup measures, upper semi-continuous functions, the Vietoris hyperspace topology, and various classes of measures.