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A Note on Equicontinuous Families of Volumes With an Application to Vector Measures

Published online by Cambridge University Press:  20 November 2018

Richard Alan Oberle
Richard Alan Oberle*
Affiliation:
The Center for Naval Analyses, Arlington, Virginia
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Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 2 , April 1974 , pp. 273 - 280
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Ando, T., Convergent sequences of finitely additive measures, Pacific J. Math. 11 (1961), 395–404.Google Scholar
2. Bessaga, C. and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces Studia Math. 18 (1958), 151–164.Google Scholar
3. Bogdanowicz, W. M. and Oberle, R. A., Topological rings of sets generated by Rickart families of contents (to appear).Google Scholar
4. Brooks, J. K., Decomposition theorems for vector measures, Proc. Amer. Math. Soc. 21 (1969), 27–29.Google Scholar
5. Brooks, J. K., On the existence of a control measure for strongly bounded vector measures, Notices Amer. Math. Soc. 18 (1971), 45.Google Scholar
5. Brooks, J. K., Qn the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001.Google Scholar
7. Brooks, J. K. and Jewett, R. S., On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298.Google Scholar
8. Day, M. M., Normed linear spaces (Springer Verlag, Berlin, 1962).Google Scholar
9. Dinculeanu, N., Vector measures, (Pergamon Press, New York, 1967).Google Scholar
10. Dunford, N. and Schwartz, J. T., Linear operators, part I (Interscience, New York, 1958).Google Scholar
11. Gould, G. G., Extensions of vector valued measures, Proc. London Math. Soc. 16 (1966), 685–704.Google Scholar
12. McArthur, C. W., On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297–302.Google Scholar
13. 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
14. Pettis, B. J., On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.Google Scholar
15. Rickart, C. E., Decompositions of additive set functions, Duke Math. J. 10 (1943), 653–665.Google Scholar
16. Robertson, A. P., On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh Sect A 68 (1969), 145–157.Google Scholar
17. Uhl, J. J., Extensions and decompositions of vector measures, J. London Math. Soc. 3 (1971), 672–676.Google Scholar