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A nonstandard proof of the Riesz representation theorem for weakly compact operators on C(Ω)

Published online by Cambridge University Press:  24 October 2008

Yeneng Sun
Yeneng Sun
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, U.S.A.
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Abstract

An easy way to construct the representing vector measures of weakly compact operators on C(Ω) is given by using the Loeb measure technique. This construction is not based on the Riesz representation theorem for linear functionals; thus we have a uniform way to treat the scalar and vector cases. Also the star finite representations of regular vector measures follow from the proof.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 141 - 145
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Albeverio, S., Fenstad, J. E., Hoeoh-Krohn, R. and Lindstrom, T. L.. Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Academic Press, 1986).Google Scholar
[2]Anderson, R. M.. Star-finite representations of measure spaces. Trans. Amer. Math. Soc. 271 (1982), 667–687.CrossRefGoogle Scholar
[3]Cutland, N. J.. Nonstandard measure theory and its applications. Bull. London Math. Soc. 15 (1983), 529–589.CrossRefGoogle Scholar
[4]Diestel, J. and Uhl, J. J.. Vector Measures. Mathematical Surveys no. 15 (American Mathematical Society, 1977).CrossRefGoogle Scholar
[5]Garling, D. J. H.. Another ‘short’ proof of the Riesz representation theorem. Math. Proc. Cambridge Philos. Soc. 99 (1986), 261–262.CrossRefGoogle Scholar
[6]Henson, C. W.. Analytic sets, Baire sets and standard part map. Canad. J. Math. 31 (1979), 663–672.CrossRefGoogle Scholar
[7]Henson, C. W. and Moore, L. C.. Nonstandard analysis and theory of Banach spaces. In Nonstandard Analysis - Recent Developments, Lecture Notes in Math. vol. 983 (Springer-Verlag, 1983), pp. 1–93.Google Scholar
[8]Hurd, A. E. and Loeb, P. A.. An Introduction to Nonstandard Real Analysis (Academic Press, 1985).Google Scholar
[9]Loeb, P. A.. Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Amer. Math. Soc. 211 (1975), 113–122.CrossRefGoogle Scholar
[10]Loeb, P. A.. Weak limits of measures and the standard part map. Proc. Amer. Math. Soc. 77 (1979), 128–135.CrossRefGoogle Scholar
[11]Loeb, P. A.. A functional approach to nonstandard measure theory. Contemp. Math. 26 (1984), 251–261.CrossRefGoogle Scholar
[12]Loeb, P. A.. Measure spaces in nonstandard models underlying standard stochastic processes. In Proc. Intern. Congr. Math. (Warsaw, 1983) (PWN Polish Scientific Publishers, 1984).Google Scholar
[13]Luxemburg, W. A. J.. A general theory of monads. In Applications of Model Theory to Algebra, Analysis, and Probability (Holt, Rinehart, and Winston, 1969), pp. 18–86.Google Scholar
[14]Osswald, H. and Sun, Y. N.. On the extensions of vector valued Loeb measures. (To appear.)Google Scholar
[15]Sun, Y. N.. On the theory of vector valued Loeb measures and integration. (To appear.)Google Scholar
[16]Zivaljevic, R. T.. Loeb completion of internal vector valued measures. Math. Scared. 56 (1985), 276–286.Google Scholar