The Convergence Determining Class of Regular Open Sets

The purpose of this paper is to prove that every sequence of closed approximable measures defined on the Borelfield of a normal topological space with values in an abelian topological group is Cauchy convergent for all Borel sets if it is Cauchy convergent for all regular open sets. In particular every sequence of measures on the Borel-field of a perfectly normal topological space which is Cauchy convergent for all regular open sets is Cauchy convergent for all Borel sets, too. 1. Preliminaries. In this paper a topological group G is always assumed to be abelian. The system of neighborhoods of the zero element of G is denoted by J^O). A sequence aneG, ne N, is Cauchy convergent iff it is Cauchy convergent with respect to the uniformity : {{ia,b)eG x G:a b eF}:Fe.F(0)}. Let iX, £T) he a topological space. The closure of a set A <= X he denoted by Ac, and its interior by int A. A set A^Xis regular open iff A='mt Ac. The system of regular open sets is denoted by &"r The function T e S~-*T* : = int Tc e 3~T has the following properties : (i) T** = T*; (ii) Je [/implies r*<= U*; (iii) TnU=0 implies T*nU* = 0; (iv) Tcr\Uc=0 implies (TvU)* = T*kjU*. Let 3S be a o--field on X and G be a topological group. A function p:38^>-G is a measure iff for all disjoint sets .4, e 38, i e N, the sequence Œ"=i i"(^¿))nefv converges to p(\J ¿6JV Af\. Let Jfc^1; a measure p : á?-*-G is ¿f-regular iff for each Ae£% and each F e ^(0) there exists Ke JÍT, K^A suchthat p(3§ r\(AK)) : = {p(B C\(AK)):Be 3$} <= F. For the case of a Banach space G, regularity in the sense of this definition is the usual regularity, defined in terms of the semivariation. Received by the editors May 3, 1971 and, in revised form, May 23, 1972. AMS (MOS) subject classifications (1970). Primary 60B10; Secondary 28A45.