Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-26T07:15:11.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniformity for weak order convergence of Riesz space-valued measures

Published online by Cambridge University Press:  17 April 2009

Jun Kawabe
Jun Kawabe
Affiliation:
Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz space-valued σ-measures.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 2 , April 2005 , pp. 265 - 274
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Aliprantis, C.D. and Burkinshaw, O., Positive operators (Academic Press, London, 1985).Google Scholar
[2]D'Aniello, E. and Wright, J.D.M., ‘Finding measures with given marginals’, Quart. J. Math. 4 (2000), 405–416.CrossRefGoogle Scholar
[3]Billingsley, P., Convergence of probability measures (John Wiley & Sons, New York, 1968).Google Scholar
[4]Boccuto, A. and Sambucini, A.R., ‘The monotone integral with respect to Riesz space-valued capacities’, Rend. Mat. Appl. (7) 16 (1996), 255–278.Google Scholar
[5]Dekiert, M., Kompaktheit, Fortsetzbarkeit und Konvergenz von Vectormaβen, (Dissertation) (University of Essen, Duisburg, 1991).Google Scholar
[6]Fremlin, D.H., Topological Riesz spaces and measure theory (Cambridge University Press, Cambridge 1974).CrossRefGoogle Scholar
[7]James, I.M., Topological and uniform spaces (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
[8]Kawabe, J., ‘The portmanteau theorem for Dedekind complete Riesz space-valued measures’, in Nonlinear analysis and convex analysis, (Takahashi, W. and Tanaka, T., Editors) (Yokohama Publishers, 2004), pp. 149–158.Google Scholar
[9]Lipecki, Z., ‘On unique extensions of positive additive set functions’, Arch. Math. 41 (1983), 71–79.CrossRefGoogle Scholar
[10]März, M. and Shortt, R.M., ‘Weak convergence of vector measures’, Publ. Math. Debrecen 45 (1994), 71–92.CrossRefGoogle Scholar
[11]Rao, R. Rang, ‘Relations between weak and uniform convergence of measures with applications’, Ann. Math. Statist. 33 (1962), 659–680.CrossRefGoogle Scholar
[12]Riečan, B. and Neubrunn, T., Integral, measure, and ordering (Kluwer, Bratislava, 1997).CrossRefGoogle Scholar
[13]Shortt, R.M., ‘Strassen's theorem for vector measures’, Proc. Amer. Math. Soc. 122 (1994), 811–820.Google Scholar
[14]Topøe, F., Topology and measure, Lecture Notes in Math. 133 (Springer-Verlag, New York, 1970).CrossRefGoogle Scholar
[15]Vakhania, N.N., Tarieladze, V.I. and Chobanyan, S.A., Probability distributions on Banach spaces (D. Reidel Publishing Company, Dordrecht, 1987).CrossRefGoogle Scholar
[16]Vulikh, B.Z., Introduction to the theory of partially ordered spaces (Wolters-Noordhoff, Groningen, 1967).Google Scholar
[17]Wright, J.D.M., ‘Stone-algebra-valued measures and integrals’, Proc. London Math. Soc. 19 (1969), 107–122.CrossRefGoogle Scholar
[18]Wright, J.D.M., ‘The measure extension problem for vector lattices’, Ann. Inst. Fourier, Grenoble 21 (1971), 65–85.CrossRefGoogle Scholar
[19]Wright, J.D.M., ‘Measures with values in a partially ordered vector space’, Proc. London Math. Soc. 25 (1972), 675–688.CrossRefGoogle Scholar
[20]Zaanen, A.C., Introduction to operator theory in Riesz spaces (Springer-Verlag, Berlin, 1997).CrossRefGoogle Scholar