Advertisement

On the Ranges of Additive Correspondences

  • Achille Basile
Conference paper

Abstract

In this work we consider a variety of finitely additive set-valued set-functions with infinite-dimensional range-spaces. The results are à la Lyapunov, namely they furnish some conditions for the range to have compact/convex closure. Some material concerning set-valued set-functions with values in semigroups is also given both because of its own interest and because it is preparatory for the topological vector space situation.

Keywords

Topological Vector Space Mathematical Economic Infinite Dimensional Space Lyapunov Theorem Continuous Linear Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Springer, Berlin, 1994.MATHGoogle Scholar
  2. 2.
    C. D. Aliprantis, K. C. Border, and O. Burkinshaw, Economies with many commodities, Journal of Economic Theory 74 (1997), 62–105.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    C. D. Aliprantis, D. J. Brown, and O. Burkinshaw, Existence and optimality of competitive equilibria, Springer, New York, 1990.CrossRefGoogle Scholar
  4. 4.
    T. E. Armstrong and K. Prikry, Lyapunov’s theorem for nonatomic, finitely additive, bounded, finite-dimensional, vector-valued measures, Trans. Amer. Math. Soc. 266 (1981), 499–514.MathSciNetMATHGoogle Scholar
  5. 5.
    T. E. Armstrong and M. K. Richter, The core-Walras equivalence, Journal of Economic Theory 33 (1984), 116–151.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    T. E. Armstrong and M. K. Richter, Existence of nonatomic core-Walras allocations, Journal of Economic Theory 38 (1986), 137–159.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A. Avallone and A. Basile, Lyapunov-Richter theorem in the finitely additive setting, Journal of Mathematical Economics 22 (1993), 557–561.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Avallone and A. Basile, Lyapunov-Richter theorem in B-convex spaces, Journal of Mathematical Economics 29 (1998).Google Scholar
  9. 9.
    W. G. Bade and P. C. Curtis, The Wedderburn decomposition of commutative Banach algebras, American Journal of Mathematics 82 (1960), 851–866.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    A. Basile, Finitely additive correspondences, Proc. Amer. Math. Soc. 121 (1994), 883–891.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    D. Blackwell, The range of certain vector integrals, Proc. Amer. Math. Soc. 2 (1951), 390–395.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    M. A. Costé, Sur les multimesures a valeurs fermees bornees d’un espace de Banach, Compte Rendue de l’Academie des Sciences de Paris 280 (1975), 567–570.MATHGoogle Scholar
  13. 13.
    M. A. Costé, Densite des selecteurs d’une multimesures a valeurs convexes fermees bornees d’un espace de Banach separable, Compte Rendue de l’Academie des Sciences de Paris 282 (1976), 967–969.MATHGoogle Scholar
  14. 14.
    J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press, 1995.Google Scholar
  15. 15.
    J. Diestel and J. J. Uhl, Vector Measures, American Mathematical Society, Providence-Rhode Island, 1977.MATHGoogle Scholar
  16. 16.
    L. Drewnowski, Additive and countably additive correspondences, Annales Societatis Mathematicae Polonae, Series I: Commentationes Mathematicae XIX (1976), 25–54.MathSciNetGoogle Scholar
  17. 17.
    A. Dvoretzky, A. Wald and J. Wolfowitz, Relations among certain ranges of vector measures, Pacific Journal of Mathematics 1 (1951), 59–74.MathSciNetMATHGoogle Scholar
  18. 18.
    V. M. Kadets, A remark on Lyapunov’s theorem on a vector measure, Functional Analysis and Applications 25 (1991), 295–297.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    V. M. Kadets and G. Shekhtman, The Lyapunov theorem for l p-valued measures, Saint Petersburg Mathematical Journal 4 (1993), 961–966.MathSciNetGoogle Scholar
  20. 20.
    M. A. Khan and N. C. Yannelis (eds.), Equilibrium Theory in Infinite Dimensional Spaces, Springer-Verlag, Berlin, 1991.MATHGoogle Scholar
  21. 21.
    A. Mas-Colell and W. R. Zame, Equilibrium theory in infinite dimensional spaces, in: Handbook of Mathematical Economics, Vol. IV, North-Holland, 1991, 1835-1898.Google Scholar
  22. 22.
    H. Richter, Verallgemeinerung eines in der Statistik benötigten Satzes der Masstheorie, Mathematische Annalen 150 (1963), 85–90.MathSciNetCrossRefGoogle Scholar
  23. 23.
    D. Schmeidler, Convexity and compactness in countably additive correspondences, in: Differential games and related topics (eds. H. W. Kuhn and G. P. Szego), North-Holland, 1971.Google Scholar
  24. 24.
    D. Schmeidler, On set correspondences into uniformly convex Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 97–101.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    C. Swartz, A generalized Orlicz-Pettis theorem and applications, Mathematische Zeitschrift 163 (1978), 283–290.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    K. Vind, Edgeworth allocations in exchange economy with many traders, International Economic Review 5 (1964), 165–177.MATHCrossRefGoogle Scholar
  27. 27.
    H. Weber, Group and vector valued s-bounded contents, in: Measure Theory Oberwolfach 1983, LNM 1089, Springer-Verlag, 1984.Google Scholar
  28. 28.
    H. Weber, Compact convex sets in non-locally convex spaces, Note di Matematica 12 (1992), 271–289.MathSciNetMATHGoogle Scholar
  29. 29.
    E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Springer-Verlag, New York, 1985.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1998

Authors and Affiliations

  • Achille Basile
    • 1
  1. 1.Facoltà di Economia, Dipartimento di Matematica e StatisticaUniversità Federico IINapoliItaly

Personalised recommendations