有界半特征的同胚定理及其应用

The Homeomorphism Theorem of Bounded Semicharacters and Its Applications

设（Ｓ，＋，ｅ）为一可交换半群，有单位元ｅ．称函数ρ：ｓ→［－１，１］为有界半特征，若ρ（ｅ）＝１且ρ（ｓ＋ｔ）＝ρ（ｓ）ρ（ｔ），ｓ，ｔ∈Ｓ．设Ｈ为一些有界半特征所成的集合，Ｍ（Ｈ）为Ｈ上的全体有限Ｒａｄｏｎ测度，则有下面的同胚定理是到ＲＳ的某个子集的同胚映射．应用同胚定理，给出了局部紧空问上的随机测度的相应的经典命题的较简单新证明，且无需第二可数性条件．

Let(S, +,e) be acommutative semigroup with neutral element e . A bounded semicharacter is afunc-tion ρ:S→[-1,1]suchthat ρ(e)=1and ρ(s+t)=ρ(s)ρ(t),s,t∈S.Let H be a set of some bounded semicharacters, M( H) be the set of all finite Radon measures on H. Then holds the homeomorphism theo-rem: The mapping μ→Lμ:= Hρ(s)μ(dρ),s∈S is a homeomorphism fromM(H) to a subset of RS. Using this theorem, a new and simpler proof is given for the corresponding canonical propositions of the random measure on a locally compact space, without the second countability condition.