Controls of families of finitely additive functions. (English) Zbl 0648.28007

Let R be a Boolean ring, let G be a complete Hausdorff Abelian topological group, and let M be a family of s-bounded \((=\) exhaustive) functions on R with values in G. The author deals with the problem of the existence of an additive function \(\gamma:R\to G\) such that the Fréchet-Nikodým topologies generated by M and \(\gamma\) coincide. Of main concern is the case where M is uniformly s-bounded. The answer is then affirmative in the following two situations: (A) G is metrizable and the elements of M are atomic; (B) G is a complete locally convex topological vector space and M is countable. The proofs are based on H. Weber’s results concerning the Boolean algebra of all s-bounded Fréchet-Nikodým topologies on R [Pac. J. Math. 110, 471-495 (1984; Zbl 0489.28008)]. Moreover, an example is given to show that the second result fails without the assumption of uniform s-boundedness.
{Reviewer’s remarks: (1) Lemma 2 follows directly from results of L. Drewnowski [Stud. Math. 50, 203-224 (1974; Zbl 0285.28015), Theorem 4.8] and H. Weber [op. cit.]. (2) In the proof of Theorem 3 an appeal to Lemma 2 is missing. Moreover, \(\tilde R\wedge a_ M\) can be finite. (3) In the construction on pp. 300-301 the following condition is missing: If \(n\neq m\) and \(x_ n\in \Omega\), then \(x_ n\neq x_ m\). (4) The background of the example mentioned above is unnecessarily complicated. Indeed, it is enough to find, given a strictly increasing sequence (\({\mathfrak m}_ i)\) of infinite cardinal numbers, a \(\sigma\)-ring R and G as above such that every measure \(\mu\) : \(R\to G\) satisfies the \({\mathfrak m}_ n\)-chain condition for some n, and for every i there exists a measure \(\mu_ i: R\to G\) which does not satisfy the \({\mathfrak m}_ i\)- chain condition. (5) Part of what the author calls “Weber’s completion principle” is due, in a more general setting, to T. G. Kiseleva [Vestn. Leningr. Univ., Ser. I 22, No.13, 51-57 (1967; Zbl 0189.016)].}
Reviewer: Z.Lipecki


28B10 Group- or semigroup-valued set functions, measures and integrals
28B05 Vector-valued set functions, measures and integrals