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On Uniform Semigroup-Valued Additive Set Functions

Published online by Cambridge University Press:  20 November 2018

Geoffrey Fox  and
Pedro Morales
Geoffrey Fox
Affiliation:
Département de Mathématiques et de Statistiques, Université de MontréalMontréal, Québec H3C 3J7
Pedro Morales
Affiliation:
Département de Mathématiques et D'Informatique, Université de SherbrookeSherbrooke, Québec J1K 2R1
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Abstract

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The main results of this paper are the following: (1) An extension theorem for a uniform semigroup-valued measure on a ring to the generated σ-ring. This result unifies the classieal Hahn-Carathéodory theorem, the extension theorem of Sion and a more recent result of Weber.

(2) A theorem stating that every monocompact additive uniform semigroup-valued set function on a semiring is σ-additive. This result generalizes several earlier theorems of Alexandroff, Dinculeanu-Kluvanek, Glicksberg, Huneycutt, Mallory, Marczewski, Millington and Topsøe.

Keywords

28B1004A15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 1 , 01 March 1983 , pp. 26 - 34
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Alexandroff, A. D., Additive set functions in abstract spaces II, Mat. Sb. 9 (51) (1941), 563-628.Google Scholar
2. Bourbaki, N., Topologie générale, 3rd. éd., Actualités Sci. Ind. No. 1143, Chap. 3 and 4, Hermann, Paris (1960).Google Scholar
3. Dinculeanu, N., Vector Measures, Pergamon Press, New York (1967).Google Scholar
4. Dinculeanu, N. and Kluvanek, I., On vector measures, Proc. London Math. Soc. 17 (1967), 505-512.Google Scholar
5. Drewnowski, L., Topological ring of sets, continuous set functions, integration III, Bull. Acad. Polon. Sci., Ser. Sci. Math., Astr., Phys. 20 (1972), 439-445.Google Scholar
6. Fox, G. and Morales, P., Uniform semigroup-valued measures I, Rapport de recherche N. 80–17, Université de Montréal (Septembre 1980), 20 pages.Google Scholar
7. Glicksberg, I., Representation of functional by integrals, Duke Math. J. 19 (1952), 253-261.Google Scholar
8. Hildebrandt, T. H., On unconditional convergence in normed vector spaces, Bull. Amer. Math. Soc. 46 (1940), 959-962.Google Scholar
9. Huneycutt, J. E. Jr, Extensions of abstract valued set functions, Trans. Amer. Math. Soc. 141 (1969), 505-513.Google Scholar
10. Lipecki, Z., Extensions of tight set functions with values in a topological group, Bull. Acad. Polon. Sci., Ser. Sci. Math., Astr., Phys. 22 (1974), 105-113.Google Scholar
11. Mallory, D., Extension of set functions to measures and applications to inverse limit measures, Canad. Math. Bull. 18 (1975), 547-553.Google Scholar
12. Marczewski, E., On compact measures, Fund. Math. 40 (1953), 113-124.Google Scholar
13. Millington, H., Products of group-valued measures, Studia Math. 54 (1975), 7-27.Google Scholar
14. Morales, P., Regularity and extension of semigroup valued Baire measures, Proc. Conf. Measure Theory Oberwolfach 1979, Lect. Notes Math. 794, Springer-Verlag, New York (1980), 317-323.Google Scholar
15. Sion, M., Outer measures with values in a topological group, Proc. London Math. Soc. 19 (1969), 89-106.Google Scholar
16. Sion, M., A theory of semigroup value measures, Lect. Notes Math. 355, Springer-Verlag, New York (1973).Google Scholar
17. Stephenson, R.M., Pseudo-compact spaces, Trans. Amer. Math. Soc. 134 (1968), 437-448.Google Scholar
18. Topsøe, F., On construction of measures, Copenhagen University Preprint Series No. 27 (1974).Google Scholar
19. Topsøe, F., Approximating pavings and construction of measures, Coll. Math. 42 (1979), 377-385.Google Scholar
20. Weber, H., Fortsetzung von Massen mit Werten in uniformen Halbgruppen, Arch. Math. 27 (1976), 412-423.Google Scholar