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The atoms of a countable sum of set functions. (English) Zbl 0665.28003
The set of atoms of a nonnegative set function which is a countable sum of nonnegative set functions is expressed. An analogous theorem for semigroup valued measures is given. Moreover it is shown that the sum of countably many atomic measures is an atomic measure.

28A10 Real- or complex-valued set functions
28A12 Contents, measures, outer measures, capacities
28B10 Group- or semigroup-valued set functions, measures and integrals
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[1] ARSTEIN Z.: Set valued measures. Trans. Amer. Math. Soc. 165, 1972, 103-121.
[2] BERBERIAN S. K.: Measure and Integration. New York 1965. · Zbl 0126.08001
[3] CAPEK P.: Théorèmes de décomposition en théorie de la mesure I. II. Publications du Séminaire d’Analyse de Brest, juin 1976.
[4] CAPEK P.: Decomposition theorems in measure theory. Math. Slovaca 31, 1981, No. 1, 53-59. · Zbl 0452.28002
[5] CAPEK P.: The pathological infìnity of measures. Suppl. ai Rendiconti del Circolo Mat. Di Palermo. Série II-6, 1984. · Zbl 0596.28001
[6] FICKER V.: On the equivalence of a countable disjoint class of sets of positive measure and a weaker condition than total \sigma -fìniteness of measures. Bull. Austral. Math. Soc. 1, 1969, 237-243. · Zbl 0174.09103
[7] GODET-THOBIE C.: Multimesures et multimesures de transitions. Thèse. Montpellier. 1985. · Zbl 0344.28003
[8] HAHN H., ROSENTHAL A.: Set Functions. The University of New Mexico Press 1948. · Zbl 0033.05301
[9] HIAI F.: Radon-Nikodym theorems for set-valued measures. Journal of Miltiv. Analysis 8, 1978, 96-118. · Zbl 0384.28006
[10] JOHNSON R. A.: Atomic and nonatomic measures. Proc. Amer. Math. Soc., 25, 1970, 650-655. · Zbl 0201.06201
[11] SIKORSKI R.: Boolean Algebras. Springer-Verlag 1969. · Zbl 0191.31505
[12] DREWNOWSKI L.: Additive and countably additive correspondences. Commentationes Mathem. XIX 1976. · Zbl 0364.28014
[13] CAPEK P.: Abstract comparison of the properties of infìnite measures. Acta Math. Univ. Comen.
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