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Weak compactness and convergences in L 1E’ [E]

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Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 3))

Abstract

Suppose that (Ω, ℱ, μ) is a complete probability space, E is a Banach space, E’ is the topological dual of E and ρ is a lifting in ℒ ∞ (μ). We state several convergences and weak compactness results in the Banach space (L 1E’ , [E], N̄ 1) of weak*-scalarly integrable E’-valued functions via the Banach space (L 1,ρE’ , [E], N̄ 1,ρ) associated to the lifting ρ.) Applications to Young measures, Mathematical Economics, Minimization problems and Set-valued integration are also presented.

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References

  1. Amrani, A.: Lemme de Fatou pour l’intégrale de Pettis. Publications Mathématiques, Spain 42, 67–79 (1998)

    Google Scholar 

  2. Amrani, A., Castaing, O., Valadier, M.: Méthodes de troncature appliquées des problèmes de convergence faible ou forte dans L 1 . Arch. Rational Mech. Anal. 117, 167–191 (1992)

    Google Scholar 

  3. Ando, T., Shintani, T.: Best approximants in L1-space. Wahrscheinlichkeistheorieverw. Gebiete 33, 33–39 (1975)

    Article  Google Scholar 

  4. Artstein, Z.: A note on Fatou’s lemma in Several dimensions. J. Math. Economies 6, 277–282 (1979)

    Article  Google Scholar 

  5. Balder, E.J.: On Prohorov’s theorem for transition probabilities. Sém. Anal. Convexe 19, 9.1–9.11 (1989)

    Google Scholar 

  6. Balder, E.J.: On equivalence of strong and weak convergence in L 1-spaces under extreme point conditions. Israel J. Math. 75, 21–47 (1991)

    Article  Google Scholar 

  7. Balder, E.J.: Lectures on Young measures, Preprint 9517 (23/03/1995) CERE-MADE, Université Paris-Dauphine, France 1995

    Google Scholar 

  8. Balder, E.J., Hess, C.: Fatou’s Lemma for multifunctions with unbounded Values. Math. Oper. Res. 20, 175–188 (1995)

    Article  Google Scholar 

  9. Balder, E.J., Hess, O.: Two generalizations of Komlós theorem with Lower Closure-Type Aplications. Journal of Convex Analysis 3, 25–44 (1996)

    Google Scholar 

  10. Benabdellah, H.: Compacité, convergences et équations d’évolution. Thèse de Doctorat d’ Etat, Université Mohammed V- Rabat 1995

    Google Scholar 

  11. Benabdellah, H., Castaing, O.: Weak compactness criteria and convergences in L 1E(μ). Collectanea Mathematica XLVIII, 423–448 (1997)

    Google Scholar 

  12. Benabdellah, H., Castaing, C.: Weak compactness and convergences in L l E (μ) . C. R. Acad. Sci. Paris, t.321 Ser. I, 165–170 (1995)

    Google Scholar 

  13. Bourbaki, N.: Intégration, Chap. 9: Intégration sur les espaces topologiques séparés. Hermann, Paris 1969

    Google Scholar 

  14. Bourgain, J.: An averaging result for l 1-sequences and applications to weakly conditionally compact sets in L l x . Israel J. of Math. 32, 289–298 (1979)

    Article  Google Scholar 

  15. Bourgain, J., Fremlin, D.H., Talagrand, M.: Pointwise compact sets of Bairemeasurable functions. Amer. J. of Math. 100(4), 845–886 (1978)

    Article  Google Scholar 

  16. Brooks, J.K., Dinculeanu, N.: Weak compactness in the space of Bochner integrable functions and applications. Adv. in Math. 24, 172–188 (1977)

    Article  Google Scholar 

  17. Castaing, O., Saadoune, M.: Dunford-Pettis types theorem and Convergences in Set-valued Integration. Journal on Nonlinear and Convex Analysis 1(1), 37–71 (2000)

    Article  Google Scholar 

  18. Castaing, O., Guessous, M.: Convergences in L 1 x (μ) . Adv. Math. Econ., 1, 17–37 (1999)

    Article  Google Scholar 

  19. Castaing, O, Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Springer-Verlag, Berlin-Heidelberg-New-York 1977

    Book  Google Scholar 

  20. Castaing, O., Valadier, M.: Weak convergence using Young measures, Functiones et Approxiatio 26, 7–17 (1998)

    Google Scholar 

  21. Diaz, S.: Weak compactness in L l (μ,X). Proc. Amer. Math. Soc. 124(9), 2685–2693 (1996)

    Article  Google Scholar 

  22. Diestel, J., Uhl Jr., J.J.: Vector measures, Math. Surveys 15. Amer. Math. Soc., Providence, RI 1977

    Google Scholar 

  23. Diestel, J., Ruess, W.M., Schachermayer, W.: Weak compactness in L1(μ, X). Proc. Amer. Math. Soc. 118 (2), 447–453 (1993)

    Google Scholar 

  24. Dudley, R.M.: Convergence of Baire measures. Studia Math. 27, 251–268 (1966)

    Google Scholar 

  25. Dudley, R.M.: Real analysis and probability. Wadsworth & Brooks/Cole Mathematics Series, California 1989

    Google Scholar 

  26. Dunford, N., Schwartz, J.T.: Linear operators, part I. Interscience, New York 1964

    Google Scholar 

  27. Edgar, G.A., Talagrand, M.: Liftings of functions with values in a completely regular space. Proc. Amer. Math. Soc. 78 no 3, 345–349 (1980)

    Article  Google Scholar 

  28. Gaposkhin, V.F.: Convergences and limit theorems for sequences of random variables. Theory of Probability Appl. 17, 379–400 (1979)

    Article  Google Scholar 

  29. Grothendieck, A.: Espaces vectoriels topologiques. Publi. de la Soc. Math. de Sao Paulo, Sao Paulo 1954 (1964)

    Google Scholar 

  30. Khan, M. Ali, Majumdar, M.: Weak sequential Compactness in L 1 (μ,X) and an Approximate Version of Fatou’s lemma. J. Math. AnaL Appl. 114, 569–573(1986)

    Article  Google Scholar 

  31. Jarchow, H.: Locally Convex Spaces. Mathematische Leitfäden. B.G. Teubner, Stuttgart 1981

    Book  Google Scholar 

  32. Levin, V.L.: Convex analysis in spaces of measurable functions and its applications in Mathematics and Economics. Nauka, Moscow, 1985 (in Russian).

    Google Scholar 

  33. Olech, C.: Existence theory in optimal control. In: Control theory and topics in functional analysis. IAEA, Vienna 1, pp.191–228, 1976

    Google Scholar 

  34. Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press, New York 1967

    Google Scholar 

  35. Rzeiuchowski, T.: Impact of dentability on weak convergence in L 1 . Bolletino U.M.I. 7, 71–80 (1992)

    Google Scholar 

  36. Rustichini, A.: A counterexample and an exact version of Fatou’s lemma in infinite dimensional spaces. Arch. Math. 52, 357–362 (1989)

    Article  Google Scholar 

  37. Schliichtermann, G.: Weak Cauchy sequences in L ∞ (μ,X) . Studia Math. 116(3), 271–281 (1995)

    Google Scholar 

  38. Schmeidler, D.: Fatou’s lemma in several dimensions. Proc. Amer. Math. Soc. 24, 300–306 (1970)

    Google Scholar 

  39. Slaby, M.: Strong convergence of vector-valued pramarts and subpramarts. Probability and Math. Stat. 5, 187–196 (1985)

    Google Scholar 

  40. Talagrand, M.: Weak Cauchy sequences in L l (E) . Amer. J. Math. 106, 703–724 (1984)

    Article  Google Scholar 

  41. Talagrand, M.: Pettis integral and measure theory. Mem. Amer. Math. Soc, no 307, vol 51 (1984)

    Google Scholar 

  42. Tulcéa, A. and C. Ionescu: Topics in the theory of lifting, Ergeb. Math. Grenzgeb. (3) 48, Spinger Verlag, New York 1969

    Book  Google Scholar 

  43. Ulger, A.: Weak compactness in L 1 (μ, X), Proc. Amer. Math. Soc. 113 n° 1, 143–150 (1991)

    Google Scholar 

  44. Valadier, M.: Différents cas où, grâce à une propriété d’extrémalité, une suite de fonctions intégrables faiblement convergente converge fortement. Séminaire d’Analyse convexe, Montpellier (1989). Exposé No 5, pp. 5–1; 5–20

    Google Scholar 

  45. Valadier, M.: Young measures. In: Methods of Nonconvex Analysis (A. Cellina ed.). Lecture Notes in Math. 1446, pp. 152–188 Springer-Verlag, Berlin 1990

    Chapter  Google Scholar 

  46. Valadier, M.: A course on Young measures. Workshop di Teoria della Misura e Analisi Reale, Grado, September 19-October 2 (1993). Rend. Matematica Trieste suppi. (26), 349–394 (1994)

    Google Scholar 

  47. Visintin, M.: Strong convergence results related to strict convexity. Comment. Partial Diff. Equation 9, 439–466 (1984)

    Google Scholar 

  48. Yannelis, N.C.: Equilibria in Non-cooperative Models of Competition. J. Economic. Theory. 41, 96–111 (1987)

    Article  Google Scholar 

  49. Yannelis, N.C.: Fatou’s lemma in Infinite Dimensional spaces. Proc. Amer. Math. Soc. 1988

    Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, B.P. S15, Marrakech, Maroc

    Houcine Benabdellah

  2. Département de Mathématiques, case 051, Université Montpellier II, F-34095, Montpellier cedex 5, France

    Charles Castaing

Authors
  1. Houcine Benabdellah
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  2. Charles Castaing
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Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, 153-0041, Meguro-ku Tokyo, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Department of Economics, Keio University, 2-15-45 Mita, 108-8345, Minato-ku Tokyo, Japan

    Toru Maruyama (Professor) (Professor)

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Benabdellah, H., Castaing, C. (2001). Weak compactness and convergences in L 1E’ [E]. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 3. Springer, Tokyo. https://doi.org/10.1007/978-4-431-67891-5_1

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  • DOI: https://doi.org/10.1007/978-4-431-67891-5_1

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-65937-2

  • Online ISBN: 978-4-431-67891-5

  • eBook Packages: Springer Book Archive

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