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