Extensions of group‐valued regular Borel measures

Abstract

In ([ill) a result is proved about the extension of regular BOREL measures. The main result isTheorem ([Ill, Theorem 10). Let X and Y be compact HAUSDORFF spaces, 9 : X -. Y a continuous onto-mapping and po a non-negative regular BOREL measure on Y. Then there exists a non-negative regular BOREL memure p on X such that p ( A ) = p 0~-1 ( A ) ) ,Theorem 9, easily reduces to this theorem.) This theorem is a simple consequence of the HAHN-BANACH theorem ([4], [lo], [9]). First we make some remarks about notations. N will always be the set of natural numbers. For a compact HAUSDORFF space X , C ( X ) will denote the set of all real-valued continuous functions on X , a ( X ) the set of all BOREL suhsets of X, B o ( X ) all bounded, real-valued BOREL measurable functions on X , B ( X ) all bounded realvalued functions on X, and llhIl=sup I h (~) j , for any hE R o ( X ) . I n a vector lattice E order-convergence, order-continuity, order o-continuity, etc., are taken in the sense of ([?I, p. 184; see also [13], [IS]). B a ( X )will denote the set of all bounded, real-valued, RAIRE measurable functions on X. A11 locally convex spaces are taken on the field of real numbers. &,(X) denotes all BAIRE subsets of X.In thjs paper we shall prove some extensions of this result. With the above notations we shall prove the following result.

On extension theorems for M-measures in l-groups

“…For the sake of simplicity, we follow the notations of [3]. Note that in the literature there exist even some extension theorems for Riesz space-valued set functions in a more general abstract context (see for instance [6], [8], [11]). …”