Polish空间上的概率测度所构成的空间

journal6 ›› 2008, Vol. 29 ›› Issue (2): 25-26.

Polish空间上的概率测度所构成的空间

（湖南财经高等专科学校，湖南长沙 410205）

出版日期:2008-03-25 发布日期:2012-05-25
作者简介:方涛(1961-)，男，湖南临湘人，湖南财经高等专科学校副教授，主要从事数学教育与函数论研究.

On Spaces of All Probabilities on Polish Spaces

(Hunan Financial and Economic College,Changsha 410205,China)

Online:2008-03-25 Published:2012-05-25

摘要/Abstract

摘要：令Ｓ为Polish空间，Ｍ1（Ｓ）为其上所有的概率构成的空间，赋予Ｍ１（Ｓ）弱拓扑.设｛Ｘｎ｝ｎ≥１为一列Ｍ１（Ｓ）列值的随机变量，｛μｎ｝ｎ≥１为相应的一阶矩测度序列，那么当ｎ→∞时，若｛μｎ｝ｎ≥１在Ｓ上是指数胎紧的，则｛Ｘｎ｝ｎ≥１在Ｍ１（Ｓ）上是指数胎紧的.此外，当Ｓ局部紧时，如下的度量诱导出Ｍ１（Ｓ）上的弱拓扑：ｄ（μ，μ－）＝ｓｕｐｆ∈Ｆ｜μ（ｆ）－μ－（ｆ）｜，ｕ,ｕ∈Ｍ１（Ｓ）．其中F是S上α－Ｈｌｄｅｒ范数不超过某正常数的有界函数全体，α∈（０，１].

关键词： 指数胎紧, 一阶矩测度, 弱拓扑, ＨOｌｄｅｒ连续

Abstract: Assume S　is a Polish space and Ｍ１(s) the space of all probabilities on it.Endow Ｍ１(s) with the weak topology.Let ｛Ｘｎ｝ｎ≥１ be a sequence of random variables Ｍ１（ｓ）-valued and ｛μｎ｝ｎ≥１　its first moment measure sequence.Then ｛Ｘｎ｝ｎ≥１ is exponentially tight on Ｍ１(s) provided so is ｛μｎ｝ｎ≥１ on S.Moreover,when S is locally compact,the weak topology on Ｍ１(s) can be induced by the following metric:ｄ（μ，μ－）＝ｓｕｐｆ∈F｜μ（ｆ）－μ（ｆ）｜μ，μ－∈Ｍ１（Ｓ），where,F is the set of bounded continuous functions on S with α-Ｈｌｄｅｒ　norm is uniformly bounded by a C>0，and α∈（０，１].

Key words: exponentially tight, first moment measure, weak topology, Ｈlder continuous

方涛. Polish空间上的概率测度所构成的空间[J]. journal6, 2008, 29(2): 25-26.

FANG Tao. On Spaces of All Probabilities on Polish Spaces[J]. journal6, 2008, 29(2): 25-26.