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Cylindrical probabilities and the differentiation of vector measures

Published online by Cambridge University Press:  22 January 2016

Brian Jefferies
Brian Jefferies*
Affiliation:
Department of Mathematics, University of Wollongong, P. O. Box 1144, Wollongong, N. S. W. 2500, Australia
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There are many results in probability theory on vector spaces which rely implicitly on the approximation of a given cylindrical probability by cylindrical probabilities with moments; for example, this is the basic idea behind the proof of the Radon equivalence of the weak and strong topologies of a metrizable space (Schwartz [13] p. 162). The technique of approximation by cylindrical measures with moments can be systematically developed. In particular, it follows that if each member of a family of cylindrical probabilities with moments is decomposable, then the limits of these cylindrical probabilities are often regular and so they are σ-additive.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 112 , December 1988 , pp. 25 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Badrikian, A., Seminaire sur les fonctions eleatoires et les mesures cylindriques, Lecture Notes in Math., 139 (1970).Google Scholar
[2] Chi, G. Y. H., On the Radon-Nikodym theorem in locally convex spaces, Lecture Notes in Math., 541 (1975), 199–210.Google Scholar
[3] Diestel, J. and Uhl, J. J. Jr., Vector Measures, Mathematical Surveys No. 15, American Math. Soc., Providence, 1977.Google Scholar
[4] Dunford, N. and Schwartz, J. R., Linear Operators, Pt I, Interscience, New York, 1958.Google Scholar
[5] Glimm, J. and Jaffe, A., Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1981.Google Scholar
[6] Jefferies, B., Collectively measurable sets and abstract Wiener spaces, Indiana Univ. Math. J. 33 (1984), 143–158.Google Scholar
[7] Jefferies, B., The variation of vector measures and cylindrical concentration, Illinois J. Math., 30 (1986), 511–526.Google Scholar
[8] Pietsch, A., Nuclear Locally Convex Spaces, Springer-Verlag, New York, 1972.Google Scholar
[9] Schachermayer, W., Mesures cylindriques sur les espaces de Banach qui ont le Radon-Nikodým, C.R. Acad. Sci. Paris, Ser. A282 (1976), 227–229.Google Scholar
[10] Schaefer, H. H., Topological Vector Spaces, Springer-Verlag, New York, 1971.Google Scholar
[11] Schwartz, L., Seminaire Laurent Schwartz, Ecole Polytechnique, exposes no 3, 4, 11, 12, 13, annee 1969-1970.Google Scholar
[12] Schwartz, L., Seminaire Maurey-Schwartz, Ecole Polytechnique, exposes no 4, 5, 6, annee 1974-1975.Google Scholar
[13] Schwartz, L., Radon Measures in Arbitrary Topological Spaces, Tata Inst, of Fund. Res., Oxford Univ. Press, 1973.Google Scholar
[14] Thomas, E., The Lebesgue-Nikodým Theorem for Vector-Valued Radon Measures, Mem. Amer. Math. Soc., no 139 (1974).Google Scholar
[15] Vakhaniya, N. N., Tarieladze, V. I. and Tortrat, A., On the continuity of random linear functionals, Theory Prob. Appl., 26 (1981), 167–173.Google Scholar