Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-18T13:15:23.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of a Class of Equicontinuous Sets of Finitely Additive Measures with an Application to Vector Valued Borel Measures

Published online by Cambridge University Press:  20 November 2018

Richard Alan Oberle
Richard Alan Oberle*
Affiliation:
The Center for Naval Analyses, Arlington, Virginia
Rights & Permissions [Opens in a new window]

Extract

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.

Let V denote a ring of subsets of an abstract space X, let R+ denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R+, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents.

A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if limnp(An) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 2 , April 1974 , pp. 281 - 290
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bartle, R. G., Dunford, N., and Schwartz, J. T., Weak compactness and vector measures, Can. J. Math. 7 (1955), 287–305.Google Scholar
2. Bogdanowicz, W. M., A generalization of the Lebesque-Bochner-Stieltjes integral and a new approach to the theory of integration, Proc. Nat. Acad. Sci. U.S.A. 58 (1965), 492–498.Google Scholar
3. Bogdanowicz, W. M., Existence and uniqueness of extensions of volumes and the operation of completion of a volume I, Proc. Japan Acad. 42 (1966), 571–575.Google Scholar
4. Bogdanowicz, W. M., An approach to the theory of Lebesque-Bochner measurable functions and the theory of measure, Math. Ann. 164 (1966), 251–269.Google Scholar
5. Bogdanowicz, W. M., An approach to the theory of integration and theory of Lebesque-Bochner measurable functions on locally compact spaces, Math. Ann. 171 (1967), 219–238.Google Scholar
6. Bogdanowicz, W. M., An approach to the theory of integration generated by Daniell functionals and representation of linear continuous functionals, Math. Ann. 173 (1967), 34–52.Google Scholar
7. Bogdanowicz, W. M. and Oberle, R. A., Topological rings of sets generated by families of Rickart contents (to appear).Google Scholar
8. Bogdanowicz, W. M. and Oberle, R. A., Uniformly Rickart families of contents on rings of sets admitting an upper complete content (to appear).Google Scholar
9. Brooks, J. K., On the existence of a control measure for strongly bounded vector measures, Notices Amer. Math. Soc. 18 (1971), 415.Google Scholar
10. Brooks, J. K., On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001.Google Scholar
11. Brooks, J. K., Weak compactness in the space of vector measures, Bull. Amer. Math. Soc. 78 (1972), 284–287.Google Scholar
12. Buck, R. C., Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104.Google Scholar
13. Dinculeanu, N., Vector measures (Pergamon Press, New York, 1967).Google Scholar
14. Dunford, N. and Schwartz, J. T., Linear operators, part I (Interscience, New York, 1958).Google Scholar
15. Halmos, P. R., Measure theory (Van Nostrand, New York, 1950).Google Scholar
16. Hewitt, E. and Ross, K. A., Abstract harmonic analysis I (Academic Press, Inc., New York, 1963). 17.I. Kluvanek, The theory of vector valued measures, part II, Mat. Casopis Sloven. Akad. Vied. 16 (1966), 76–81 (Russian).Google Scholar
18. McArthur, C. W., On a theorem of Orlicz-Pettis, Pacific J. Math. 22 (1967), 292–302.Google Scholar
19. Oberle, R. A., Theory of a class of vector measures on topological rings of sets and generalizations of the Vitali-Hahn-Saks theorem, Ph.D. Thesis, The Catholic University of America, 1971.Google Scholar
20. Ohba, S., Decompositions of vector measures, The Reports of the Faculty of Technology, Kanagawa University, No. 10 (1972).Google Scholar
21. Peressini, A. L., Ordered topological vector spaces (Harper and Row Co., New York, 1967).Google Scholar
22. Rickart, C. E., Decompositions of additive set functions, Duke Math. J. 10 (1943), 653–665.Google Scholar
23. Robertson, A. P., On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh Sect. A 68 (1969), 145–147.Google Scholar
24. Uhl, J. J., Extensions and decompositions of vector measures, J. London Math. Soc. 8 (1971), 672–676.Google Scholar