Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T02:05:23.374Z Has data issue: false hasContentIssue false

Note on Continuous and Purely Finitely Additive Set Functions

Published online by Cambridge University Press:  20 November 2018

Wilfried Siebe*
Affiliation:
University of Bonn Department of Economics Adenauer Allee 24-42 D - 5300 Bonn 1 Federal Republic Of Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Sobczyk-Hammer respectively Yosida-Hewitt decomposition ([17], [19]) generates the class of continuous respectively purely finitely additive charges. In this paper, attention is limited to hereditable properties for these classes. It is proved that the property of continuity is preserved with respect to extensions and that if all extensions of a charge to a charge on a given field are continuous, then the original charge is continuous. An analogous heredity theorem for purely finite additivity holds true in the monogenic case.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bauer, H., Probability theory and elements of measure theory, Holt, Rinehart and Winston, 1972.Google Scholar
2. Bhaskara Rao, K. P. S. and Bhaskara Rao, M., A remark on nonatomic measures, The Annals of Mathematical Statistics 43 (1972), pp. 369370. 1380.Google Scholar
3. Bhaskara Rao, K. P. S. and Bhaskara Rao, M., Charges on Boolean Algebras and almost discrete spaces, Mathematika 20 (1973), pp. 214223.Google Scholar
4. Billingsley, P., Convergence of probability measures, John Wiley and Sons, New York-London-Sydney-Toronto, 1968.Google Scholar
5. Dubins, L. and Freedman, D., Measurable sets of measures, Pacific Journal of Mathematics 14 (1964), pp. 12111222.Google Scholar
6. Dunford, N. and Schwartz, J. T., Linear operators. Part I: General theory, Interscience, New York, 1958.Google Scholar
7. Johnson, R. A., Atomic and nonatomic measures, Proceedings of the American Mathematical Society 25 (1970), pp. 650655.Google Scholar
8. Landers, D. and Rogge, L., On the extension problem for measures, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 30 (1974), pp. 167169.Google Scholar
9. Łós, J. and Marczewski, E., Extensions of measure, Fundamenta Mathematicae 36 (1949) pp. 267 —276.Google Scholar
10. Maharam, D., Finitely additive measures on the integers, Sankhyä: The Indian Journal of Statistics, Series A: 38 (1976), pp. 4459.Google Scholar
11. Maserick, P. H., Differentiation of Lp-functions on semilattices, Københavns Universitet, Matematisk Institut, Preprint Series No. 12 (1979).Google Scholar
12. Olejček, V., Darboux property of finitely additive measure on δ-ring, Mathematica Slovaca 2, 27 (1977), pp. 195201.Google Scholar
13. Parthasarathy, K. R., Probability measures on metric spaces, Academic Press, New York-San Francisco-London, 1967.Google Scholar
14. Plachky, D., Extremal and monogenic additive set functions, Proceedings of the American Mathematical Society 54(1976), pp. 193196.Google Scholar
15. Siebe, W., Vererbbarkeitsuntersuchungen bei Inhalten und Maβen, Ph.D. Thesis, University of Miinster, Munster 1979.Google Scholar
16. Siebe, W., On the Sobczyk-Hammer decomposition of additive set functions, Proceedings of the American Mathematical Society 86(1982), pp. 447450.Google Scholar
17. Sobczyk, A. and Hammer, P. C., A decomposition of additive set functions, Duke Mathematical Journal 11 (1944), pp. 839846.Google Scholar
18. Ulam, S. M., Zur Maβtheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae 16 (1930), pp. 141150.Google Scholar
19. Yosida, K. and Hewitt, E., Finitely additive measures, Transactions of the American Mathematical Society 72 (1952), pp. 4666.Google Scholar