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Integration with respect to vector valued Radon polymeasures

Part of: Set functions, measures and integrals with values in abstract spaces Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  09 April 2009

Brian Jefferies  and
Werner J. Ricker
Brian Jefferies
Affiliation:
School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, N.S.W., Australia
Werner J. Ricker
Affiliation:
School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, N.S.W., Australia
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Abstract

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Problems dealing with certain functional calculi for systems of non-commuting operators, and ordered calculi for systems of certain types of pseudo-differential operators, can sometimes be treated via the methods of integration with respect to polymeasures. The polymeasures arising in this fashion (called Radon polymeasures) often have additional structure not available in the general theory. This allows for a more extensive class of “integrable” functions than just the product functions allowed in the abstract theory. The purpose here is to further develop special aspects of integration with respect to Radon polymeasures with a particular emphasis on identifying large classes of “integrable” functions.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 1 , February 1994 , pp. 17 - 40
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Anderson, R. F. V., ‘The Weyl functional calculus’, J. Fund. Anal. 4 (1969), 240–267.CrossRefGoogle Scholar
[2]Dellacherie, C. and Meyer, P. A., ‘Probabilités et Potential’, volume 1372 of Actualités Sci. Indust. (Herman, Paris, 1975).Google Scholar
[3]Diestel, J. and Uhl, J. J. Jr, Vector measures, Math. Surveys Monographs 15 (Amer. Math. Soc., Providence, 1977).CrossRefGoogle Scholar
[4]I. Dobrakov, ‘Feynman type integrals as multilinear integrals’, preprint.Google Scholar
[5]Dobrakov, I., ‘On integration in Banach spaces VIII (Polymeasures)’, Czechoslovak Math. J. 37 (1987), 487–506.CrossRefGoogle Scholar
[6]Dobrakov, I., ‘Representation of multilinear operators on ×C 0(Ti)’, Czechoslovak Math. J. 39 1989, 288–302.CrossRefGoogle Scholar
[7]Dobrakov, I., ‘On integration in Banach spaces XI (Integration with respect to polymeasures)’, Czechoslovak Math. J. 40 (1990), 8–24.CrossRefGoogle Scholar
[8]Dobrakov, I., ‘On integration in Banach spaces XII (Integration with respect to polymeasures)’, Czechoslovak Math. J. 40 (1990), 424–440.CrossRefGoogle Scholar
[9]Gilbert, J. E., Ito, T. and Schreiber, B. M., ‘Bimeasure algebras and locally compact groups’, J. Funct. Anal. 64 (1985), 134–162.CrossRefGoogle Scholar
[10]Gillespie, T. A., ‘Strongly closed bounded Boolean algebras of projections’, Glasgow Math. J. 22 (1981), 73–75.CrossRefGoogle Scholar
[11]Graham, C. C. and Schreiber, B. M., ‘Bimeasure algebras on LCA groups’, Pacific J. Math. 115 (1984), 91–127.CrossRefGoogle Scholar
[12]Graham, C. C. and Schreiber, B. M., ‘Projections in spaces of bimeasures’, Canad. Math. Bull. 31 (1988), 19–25.CrossRefGoogle Scholar
[13]Jefferies, B., ‘Radon polymeasures’, Bull. Austral. Math. Soc. 32 (1985), 207–215.CrossRefGoogle Scholar
[14]Jefferies, B., ‘Summability in spaces of set functions’, in: Proceedings of the Ramanujan internat. symp. on analysis (Poona, ndia, 1987) pp. 360–372.Google Scholar
[15]Jefferies, B. and Ricker, W. J., ‘An integral calculus for systems of noncommuting operators’, in preparation.Google Scholar
[16]Kalton, N. J., ‘Exhaustive operators and vector measures’, Proc. Edinburgh Math. Soc. 19 (1975), 291–300.CrossRefGoogle Scholar
[17]Kluváknek, I., ‘Remarks on bimeasures’, Proc. Amer. Math. Soc. 81 (1981), 233–239.CrossRefGoogle Scholar
[18]Kluváknek, I. and Knowles, G., Vector measures and control systems (North Holland, Amsterdam, 1976).Google Scholar
[19]Morse, M. and Transue, W., ‘C-bimeasures and their integral extensions’, Ann. of Math. 64 (1956), 480–504.CrossRefGoogle Scholar
[20]Rao, M. M. and Chang, D. K., ‘Bimeasures and non-stationary processes’, in: Real and stochastic analysis (ed. Rao, M. M.) (Wiley, New York, 1986) pp. 7–118.Google Scholar
[21]Schwartz, L., Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute publications (Oxford University Press, Bombay, 1973).Google Scholar
[22]Treves, F., Topological vector spaces, distributions and kernels (Academic Press, New York, 1967).Google Scholar
[23]Ylinen, K., ‘Fourier transforms of noncomniutative analogues of vector measures and bimeasures with applications to stochastic processes’, Ann. Acad. Sci. Fenn. Ser. AI Math. 1 (1975), 355–385.Google Scholar
[24]Ylinen, K., ‘On vector bimeasures’, Ann. Mat. Pura Appl. (4) 117 (1978), 115–138.CrossRefGoogle Scholar