Hostname: page-component-6bb9c88b65-9c7xm Total loading time: 0 Render date: 2025-07-17T18:40:02.979Z Has data issue: false hasContentIssue false
  • English
  • Français

Joint continuity of injective tensor products of vector measures in Banach lattices

Part of: Classical measure theory Set functions, measures and integrals with values in abstract spaces Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Jun Kawabe
Jun Kawabe
Affiliation:
Department of Mathematics Faculty of Engineering Shinshu University4-17-1 Wakasato Nagano 380-8553Japan e-mail: jkawabe@gipwc.shinshu-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

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.

It is shown that the injective tensor product of positive vector measures in certain Banach lattices is jointly continuous with respect to the weak convergence of vector measures. This result is obtained by a diagonal convergence theorem for injective tensor integrals. Our approach to this problem is based on Bartle's bilinear integration theory.

Keywords

positive vector measureinjective tensor product of vector measuresweak convergence of vector measuresBanach latticeinjective tensor integral.

MSC classification

Secondary: 28A33: Spaces of measures, convergence of measures 28B05: Vector-valued set functions, measures and integrals 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A32: Spaces of linear operators; topological tensor products; approximation properties

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 2 , April 2003 , pp. 185 - 200
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bartle, R. G., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
[2]Bartle, R. G., Dunford, N. and Schwartz, J., ‘Weak compactness and vector measures’, Canad. J. Math. 7 (1955), 289–305.Google Scholar
[3]Billingsley, P., Convergence of probability measures (Wiley, New York, 1968).Google Scholar
[4]Bourbaki, N., Topologie générale, 2nd edition (Hermann, Paris, 1958).Google Scholar
[5]Csiszár, I., ‘On the weak* continuity of convolution in a convolution algebra over an arbitrary topological group’, Studia Sci. Math. Hungar. 6 (1971), 27–40.Google Scholar
[6]Dekiert, M., Kompaktheit, Fortsetzbarkeit und Konvergenz von Vectormaβen (Dissertation, University of Essen, 1991).Google Scholar
[7]Diestel, J. and Uhl, J. J. Jr, Vector measures, Amer. Math. Soc. Surveys 15 (Amer. Math. Soc., Providence, 1977).CrossRefGoogle Scholar
[8]Duchoň, M. and Kluv´nek, I., ‘Inductive tensor product of vector-valued measures’, Mat. Časopis Sloven. Akad. Vied. 17 (1967), 108–112.Google Scholar
[9]Dudley, R. M., Real analysis and probability (Wadsworth /Cole, California, 1989).Google Scholar
[10]Dunford, N. and Schwartz, J. T., Linear operators. Part 1: General theory (Wiley, New York, 1958).Google Scholar
[11]Freniche, F. J. and García-Vázquez, J. C., ‘The Bartle bilinear integration and Carleman operators’, J. Math. Anal. Appl. 240 (1999), 324–339.Google Scholar
[12]Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (Amer. Math. Soc., Providence, 1955).CrossRefGoogle Scholar
[13]James, I. M., Topological and uniform spaces (Springer, New York, 1987).Google Scholar
[14]Kawabe, J., ‘Weak convergence of tensor products of vector measures with values in nuclear spaces’, Bull. Austral. Math. Soc. 59 (1999), 449–458.Google Scholar
[15]Kawabe, J., ‘A type of Strassen's theorem for positive vector measures with values in dual spaces’, Proc. Amer. Math. Soc. 128 (2000), 3291–3300.Google Scholar
[16]Kluvánek, I., ‘On the product of vector measures’, J. Austral. Math. Soc. 15 (1973), 22–26.Google Scholar
[17]Kluvánek, I. and Knowles, G., Vector measures and control systems (North Holland, Amsterdam, 1975).Google Scholar
[18]Lewis, D. R., ‘Integration with respect to vector measures’, Pacific J. Math. 33 (1970), 157–165.CrossRefGoogle Scholar
[19]März, M. and Shortt, R. M., ‘Weak convergence of vector measures’, Publ. Math. Debrecen 45 (1994), 71–92.Google Scholar
[20]Schaefer, H. H., Banach lattices and positive operators (Springer, Berlin, 1974).Google Scholar
[21]Schwartz, L., Radon measures on arbitrary topological spaces and cylindrical measures (Tata Institute of Fundamental Research, Oxford University Press, 1973).Google Scholar
[22]Shortt, R. M., ‘Strassen's theorem for vector measures’, Proc. Amer. Math. Soc. 122 (1994), 811–820.Google Scholar
[23]Topsøe, F., Topology and measure, Lecture Notes in Math. 133 (Springer, New York, 1970).Google Scholar
[24]Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. A., Probability distributions on Banach spaces (D. Reidel Publishing Company, 1987).Google Scholar
[25]Väth, M., Volterra and integral equations of vector functions (Marcel Dekker, New York, 2000).Google Scholar
[26]White, A. J., ‘Convolution of vector measures’, Proc. Roy. Soc. Edinburgh Sect. A 73 (1974/1975), 117–135.CrossRefGoogle Scholar