Algebraic ramifications of the common extension problem for group-valued measures

Let G be an Abelian group and let μ : A → G and ν : B → G be finitely additive measures (charges) defined on fields A and B of subsets of a set X. It is assumed that μ and ν agree on A∩B, i.e. they are consistent. The existence of common extensions of μ and ν is investigated, and conditions on A and B facilitating such extensions are given. 0. Introduction. We consider the following problem: Let G be an Abelian group and let μ and ν be G-valued charges (i.e. finitely additive measures) defined on fields A and B of subsets of a set X. When does there exist a common extension of μ and ν to a charge % defined on A ∨ B, the field generated by A ∪ B? Clearly, one must assume at least that the charges μ and ν are consistent, i.e. μ = ν on A ∩ B. Earlier work of K. M. Rangaswamy and J. D. Reid [11] and K. P. S. Bhaskara Rao and R. M. Shortt [4] has shown that the answer is in the affirmative so long as G is the homomorphic image of a compact group (i.e. is cotorsion). In fact, [4] and [11] demonstrate that this property characterizes the class of cotorsion groups. In the present article, the focus shifts to include consideration of the fields A and B. In §2, an invariant d of the pair (A,B) is introduced: d is a distance function on the Stone space of A ∨ B. The function d provides information about the geometry of A and B and the algebraic structure of the ring of simple functions measurable for A∨ B. See Lemmas 2.1 and 2.3 together with Theorem 2.4. In §3, the case where A and B are isomorphic to power set algebras is considered in terms of the common extension of charges (Lemma 3.1), and 1991 Mathematics Subject Classification: 28B10, 20K15, 20K20. Work of the second author partially supported by the Fulbright Scholar Program and the Graduiertenkolleg at the University of Essen.