On the range of finitely additive functions
The authors generalize some theorems of D. Landers and C. Constantinescu concerning the range of a group-valued measure. The main result of the paper under review [Teorema I) reads, in a modified form, as follows: Let \({\mathcal R}\) be an algebra of subsets of a set S, let G be a commutative Hausdorff topological group and let \(\mu\) : \({\mathcal R}\to G\) be an additive set function with metrizable range. Then \(\mu\) is continuous (i.e., given a neighbourhood V of O in G, there exists a partition \(\{S_ 1,...,S_ n\}\subset {\mathcal R}\) of S with \(\mu (X\cap S_ i)\in V\) for all \(X\in {\mathcal R}\) and \(i=1,...,n)\) if and only if there exists a dense subset I of [0,1] and an increasing function \(f: I\to {\mathcal R}\) such that: \(1)\quad 0,1\in I\) and \(f(O)=\emptyset\) and \(f(1)=S\); 2) the functions \(I\ni t\to \mu (Y\cap f(t)),\) where \(Y\in {\mathcal R}\), form an equicontinuous family. \{Reviewer's remarks: (1) On p. 206, line 6 from above, the index ``\((p-1)2^ n+i\)'' should be ''i''. On p. 208, line 5 from above, ``\(0<p<p'<1/2^{k_ n}\)'' should be ``\(0<p<p'\) and \(p'-p<1/2^{k_ n+1}\)''. (2) For related results see \textit{A. Martellotti} [J. Math. Anal. Appl. 110, 411-424 (1985; Zbl 0616.28002)].
scientific article; zbMATH DE number 4000273
scientific article