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Decompositions of vector measures in Riesz spaces and Banach lattices

Published online by Cambridge University Press:  20 January 2009

Klaus D. Schmidt
Klaus D. Schmidt
Affiliation:
Seminar für Statistik, Universität Mannheim, A5, 6800 Mannheim, West Germany
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The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 23 - 39
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Bilyeu, R. C. and Lewis, P. W., Orthogonality and the Hewitt-Yosida theorem in spaces of measures, Rocky Mountain J. Math. 7 (1977), 629–638.CrossRefGoogle Scholar
2.Iglesias, M. Congost, Medidas y probabilidades en estructuras ordenadas, Stochastica 5 (1981), 45–68.Google Scholar
3.Diestel, J. and Faires, B., On vector measures, Trans. Amer. Math. Soc. 198 (1974), 253–271.Google Scholar
4.Diestel, J. and Uhl, J. J. Jr., Vector Measures (Providence, Rhode Island: Amer. Math. Soc. 1977).Google Scholar
5.Faires, B. and Morrison, T. J., The Jordan decomposition of vector-valued measures, Proc. Amer. Math. Soc. 60 (1976), 139–143.CrossRefGoogle Scholar
6.Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I (Amsterdam-London: North-Holland, 1971).Google Scholar
7.Niculescu, C. P., Jordan decomposition and locally absolutely continuous operators, Revue Roumaine Math. Pures Appl. 21 (1976), 343–352.Google Scholar
8.Pavlakos, P. K., The Lebesgue decomposition theorem for partially ordered semigroupvalued measures, Proc. Amer. Math. Soc. 71 (1978), 207–211.CrossRefGoogle Scholar
9.Rickart, C. E., Decomposition of additive set functions, Duke Math. J. 10 (1943), 653–665.Google Scholar
10.Schaefer, H. H., Banach Lattices and Positive Operators (Berlin-Heidelberg-New York: Springer, 1974).CrossRefGoogle Scholar
11.Schmidt, K. D., On the Jordan decomposition for vector measures, Probability in Banach Spaces IV(Lecture Notes in Mathematics, vol. 990, Berlin-Heidelberg-New York: Springer, 1983), 198–203.CrossRefGoogle Scholar
12.Schmidt, K. D., Amarts—a measure theoretic approach, Amarts and Set Function Processes (Lecture Notes in Mathematics, vol. 1042, Berlin-Heidelberg-New York: Springer, 1983), 51–236.CrossRefGoogle Scholar
13.Talagrand, M., Quand l'espace des mesures à variation bornée est-il faiblement séquentiellement complet? Proc. Amer. Math. Soc. 90 (1984), 285–288.Google Scholar
14.Uhl, J. J. Jr., Extensions and decompositions of vector measures, J. London Math. Soc. (2) 3 (1971), 672–676.CrossRefGoogle Scholar