The convergence determining class of regular open sets

Abstract

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 5i7(O). A sequence an cG, n c N, is Cauchy convergent iff it is Cauchy convergent with respect to the uniformity: {{(a, b) E G x G:a-b E F}:Fe Y (O)}. Let (X, Y7) be a topological space. The closure of a set A c X be denoted by AC, and its interior by int A. A set A c X is regular open iff A=int Ac. The system of regular open sets is denoted by ST. The function T E -T*: int Tc c ST has the following properties: (i) T* * = T*; (ii) Tc: U implies T* c U* (iii) Tr) U= 0 implies T* rn U* = 0; (iv) Tc rn Uc= 0 implies (Tu U)* = T* u U*. Let X be a a-field on X and G be a topological group. A function It: --G is a measure iff for all disjoint sets Ai E M, i E N, the sequence (n=1 [L(Ai))neN converges to [L(UieN Ai). Let Mc X; a measure It: 4--G is Sr-regular iff for each A E X and each F E Y(O) there exists K E , Kc A such that f( r, (A K)): = {(B n (A K)):B E c 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.