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1994 | 111 | 3 | 241-262
Tytuł artykułu

From weak to strong types of $L^{1}_{E}$-convergence by the Bocce criterion

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EN
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EN
Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $ℒ^{1}_{E}$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $ℒ^{1}_{ℝ}$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $ℒ^{1}_{E}$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $ℒ^{1}_{E}$ are also studied.
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Twórcy
  • Mathematical Institute, University of Utrecht, P.O. Box 80.010, 3508 Ta Utrecht, The Netherlands, balder@math.ruu.nl
Bibliografia
  • [ACV] A. Amrani, C. Castaing et M. Valadier, Méthodes de troncature appliquées à des problèmes de convergence faible ou forte dans $L^1$, Arch. Rational Mech. Anal. 117 (1992), 167-191.
  • [B1] E. J. Balder, On weak convergence implying strong convergence in $L_1$-spaces, Bull. Austral. Math. Soc. 33 (1986), 363-368.
  • [B2] E. J. Balder, On equivalence of strong and weak convergence in $L_1$-spaces under extreme point conditions, Israel J. Math. 75 (1991), 1-23.
  • [B3] E. J. Balder, From weak to strong $L_1$-convergence by an oscillation restriction criterion of BMO type, preprint No. 666, Dept. of Math., University of Utrecht, 1991.
  • [B4] E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim. 22 (1984), 570-598.
  • [B5] E. J. Balder, On Prohorov's theorem for transition probabilities, Sém. Anal. Convexe 19 (1989), 9.1-9.11.
  • [BH1] J. Batt and W. Hiermeyer, Weak compactness in the space of Bochner integrable functions, unpublished manuscript, 1980.
  • [BH2] J. Batt and W. Hiermeyer, On compactness in $L_p(μ,X)$ in the weak topology and in the topology $σ(L_p(μ,X),L_q(μ,X'))$, Math. Z. 182 (1983), 409-423.
  • [BS] J. Batt and G. Schlüchtermann, Eberlein compacts in $L_1(X)$, Studia Math. 83 (1986), 239-250.
  • [BD] J. K. Brooks and N. Dinculeanu, Weak compactness in spaces of Bochner integrable functions, Adv. in Math. 24 (1977), 172-188.
  • [C1] C. Castaing, Un résultat de compacité lié à la propriété des ensembles Dunford-Pettis dans $L^1_F(Ω,A,μ)$, Sém. Anal. Convexe 9 (1979), 17.1-17.7.
  • [C2] C. Castaing, Sur la décomposition de Slaby. Applications aux problèmes de convergences en probabilités. Economie mathématique. Théorie du contrôle. Minimisation, Sém. Anal. Convexe 19 (1989), 3.1-3.35.
  • [CV] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin, 1977.
  • [Da] B. Dawson, Convergence of conditional expectation operators and the compact range property, Ph.D. dissertation, University of North Texas, 1992.
  • [D1] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, New York, 1984.
  • [D2] J. Diestel, Uniform integrability: an introduction, School on Measure Theory and Real Analysis, Grado (Italy), October 14-25, 1991, Rend. Istit. Mat. Univ. Trieste 23 (1991), 41-80.
  • [DU] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977.
  • [DG] S. J. Dilworth and M. Girardi, Bochner vs. Pettis norms: examples and results, in: Banach Spaces, Bor-Luh Lin and W. B. Johnson (eds.), Contemp. Math. 144, Amer. Math. Soc., Providence, R.I., 1993, 69-80.
  • [Ga] V. F. Gaposhkin, Convergence and limit theorems for sequences of random variables, Theory Probab. Appl. 17 (1972), 379-400.
  • [G1] M. Girardi, Compactness in $L_1$, Dunford-Pettis operators, geometry of Banach spaces, Proc. Amer. Math. Soc. 111 (1991), 767-777.
  • [G2] M. Girardi, Weak vs. norm compactness in $L_1$, the Bocce criterion, Studia Math. 98 (1991), 95-97.
  • [HU] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182.
  • [IT] A. and C. Ionescu-Tulcea, Topics in the Theory of Lifting, Springer, Berlin, 1969.
  • [J] V. Jalby, Contribution aux problèmes de convergence des fonctions vectorielles et des intégrales fonctionnelles, Thèse de Doctorat, Université Montpellier II, 1993.
  • [Jaw] A. Jawhar, Mesures de transition et applications, Sém. Anal. Convexe 14 (1984), 13.1-13.62.
  • [N] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965.
  • [P] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
  • [SW] G. Schlüchtermann and R. F. Wheeler, On strongly WCG Banach spaces, Math. Z. 199 (1988), 387-398.
  • [S] L. Schwartz, Radon Measures, Oxford University Press, London, 1973.
  • [T] M. Talagrand, Weak Cauchy sequences in $L_1(E)$, Amer. J. Math. 106 (1984), 703-724.
  • [V1] M. Valadier, Young measures, in: Methods of Nonconvex Analysis, A. Cellina (ed.), Lecture Notes in Math. 1446, Springer, Berlin, 1990, 152-188.
  • [V2] M. Valadier, Oscillations et compacité forte dans $L_1$, Sém. Anal. Convexe 21 (1991), 7.1-7.10.
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Bibliografia
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