Polish空间上的概率测度所构成的空间

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].

On Spaces of All Probabilities on Polish Spaces

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].