Topological properties of the range of a group-valued finitely additive measure
This paper can be considered as the last link of a sequence of papers investigating the topological structure of the range of a finitely additive measure (f.a.m.), and more particularly compactness and connectedness properties. In this paper topological group-valued f.a.m.'s are taken into consideration, examining the possibility of transposing well-known results concerning group-valued countably additive measures also to the finitely additive case. Concerning relative compactness results, it is proved that Stone-type extension techniques, which can be carried out even for group valued f.a.m.'s, are less useful in this setting than in the Banach space valued f.a.m.'s one. Besides the case of dominated f.a.m.'s, the study of range connectedness is also presented for set functions satisfying less general hypotheses than the classical non-atomicity. In order to do that, different extensions of the definition of continuity for real valued f.a.m.'s, and its equivalent formulations are investigated: finally, as in the topological group setting the equivalence between continuity and semiconvexity fails to be true, two different connectedness theorems are given under the two possible assumptions. The paper also contains some examples which point out the necessity of some of the changes and arrangements that are required if we move to finitely additive set functions. The author wishes to stress that in Example 3.5, setting \(\Omega ={\mathbb{R}},\) m is not well defined; therefore it should be replaced by, say, \(=0,1.\) The author takes this opportunity to thank Prof. Z. Lipecki for this last and other remarks, and for his kind interest.
group-valued countably additive measures
scientific article; zbMATH DE number 3998132
scientific article