On the range of finitely additive functions
type: Item
instance of: scholarly article
title: On the range of finitely additive functions
zbMATH Open document ID: 0617.28007
DOI: 10.1007/BF02844731
author: E. P. de Lucia
author: Anna Bruna D'Andrea de Lucia
publication date: 1986-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 28B10
zbMATH DE Number: 4000273
zbMATH Keywords: arcwise connected
zbMATH Keywords: range of a group-valued measure
zbMATH Keywords: commutative Hausdorff topological group
zbMATH Keywords: additive set function
zbMATH Keywords: metrizable range
reviewed by: Zbigniew Lipecki
MaRDI profile type: Publication
cites work: Q3663565
cites work: On strongly continuous functions with values in a topological group
cites work: Atoms of group valued measures
cites work: The range of atomless group valued measures
cites work: Teoremi di decomposizione per funzioni finitamente additive a valori in un gruppo topologico
cites work: On control submeasures and measures
cites work: Q3950771
cites work: A characterization of group-valued measures satisfying the countable chain condition
cites work: Der Wertebereich atomloser Inhalte
cites work: Die atomare Struktur topologischer Boolescher Ringe und s-beschränkter Inhalte
full work available at URL: https://doi.org/10.1007/bf02844731
OpenAlex ID: W1967738013
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4000273