Summary
We present new modes of convergences for bounded sequences in the space L 1X (μ) of Bochner integrable functions over a complete probability space (Ω, F, μ) with values in Banach space X via the convergence of its truncated subsequences as well as we give several characterizations of weak compactness and conditionally weak compactness in L 1X (μ). New results involving subsets in L 1X (μ) which are closed in measure are obtained and also the characterizations of the Banach space X in terms of these modes of convergence.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amrani, A., Castaing C.: Weak compactness in Pettis integration. Bulletin Polish Acad. Sc. 45, No2, 139–150 (1997)
Amrani, A., Castaing, C., Valadier, M.: Méhodes de troncature appliquées à des problèmes de convergence faible ou forte dans L 1. Arch. Rational Mech. Anal. 117, 167–191 (1992)
Balder, E. J.: Infinite-dimensional extension of a theorem of Komlös. Probab. Theory Related fields 81, 185–188 (1989)
Balder, E. J.: New sequential compactness results for spaces of scarlarly integrate functions. J. M. A. A 151, 1–16 (1990)
Balder, E. J.: On Prohorov’s theorem for transition probabilities. Sern. Anal. Convexe 19, 9.1–9.11 (1989)
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)
Balder, E. J., Hess C.: Two generalizations of Komlo’s theorem with lower closure-type applications. Journal of Convex Analysis 3 (1), 25–44 (1996)
Beer, G.: Topologies on closed and closed convex subsets and the Effros measurability of set valued functions. Sém. Anal. Convexe Montpellier 2, 2.1–2.44 (1991)
Benabdellah, H., Castaing, C.: Weak compactness and convergences in L 1 E (μ). C.R. Acad. Sci. Paris 321, 165–170 (1995)
Benabdellah, H., Castaing, C.: Weak compactness criteria and convergences in L l E (n). Collectanea Mathematica XLVIII, 423–448 (1997)
Benabdellah, H., Castaing, C.: Weak compactness and convergences in L 1 E’ /[E]. Université Montpellier II, 1996, Preprint 31 pages.
Bourgain, J.: The Komlös theorem for vector valued functions. Wrije Univer-siteit Brussel (1979) (9 pages). Unpublished.
Bukhvalov, A.V.: Optimization without compactness, and Its applications. In:Operator theory: Advances and Applications Vol 75, pp.95–112. Birkhäuser Verlag 1995
Bukhvalov, A.V., Lozanovskii, G.Ya.: On sets closed with respect to convergence in measure in spaces of measurable functions. Dokl. Akad. Nauk SSSR 212, 1273–1275 (1973); Englis transi. Soviet Math. Dokl. 1563–1565 (1973)
Castaing, C.: Quelques résultats de convergence des suites adaptées. Sém. Anal. Convexe Montpellier 17, 2.1–2.24 (1987)
Castaing, C.: Méthodes de compacité et de décomposition, Applications: Minimisation, convergence des martingales, lemme de Fatou multivoque. Ann. Mat. Pura Appli. 164, 51–75 (1993)
Castaing, C.: Weak compactness and convergences in Bochner and Pettis integration. Vietnam Journal of Math. 24 (3), 241–286 (1996)
Castaing, C., Clauzure, P.: Compacité faible dans l’espace L l E et dans l’espace des multifonctions intégrablement bornées et minimisation. Ann. Mat. Pura Appl. 4 (140), 345–364 (1985)
Castaing, C., Ezzaki, F.: Convergences for convex weakly compact random sets in B-convex reflexive Banach spaces, Supplemento al Vol. XLVI, 123–149 (1998) Atti Sem. Mat. Univ. Modena
Castaing, C., Valadier, M.: Convex Analysis and Measurable multifunctions. Lecture Notes in Mathematics 580, Springer 1977
Chatterji, S.D.: A subsequence principle in probability theory. Jber.d. Dt. Math.-Verein 87, 91–107 (1985)
Diestel, J.: Geometry of Banach Spaces, Selected topics. Lectures Notes in Mathematics 485, Springer 1975
Diestel, J., Ruess, W.M., Schachermeyer, W.: Weak compactness in L l (n,X). Proc. Amer. Math. Soc. 118 (2), 447–453 (1993)
Díaz, S.: Weak compactness in L l (n,X), Proc. Amer. Math. Soc. 124 (9), 2685–2693 (1996)
Gaposkhin, V. F.: Convergence and limit theorems for sequences of random variables. Theory Probab. Appl. 17, 379–400 (1972)
Garling, D.J.H.: Subsequence principles for vector-valued random variables. Math. Proc. Camb. Phil. Soc. 86, 301–311 (1979)
Guessous, M.: An elementary proof of Komlós-Revész theorem in Hilbert spaces. Journal of Convex Analysis 4, 321–332 (1997)
Kadec, M.I., Pelczynski, A.: Bases, lacunary sequences and complemented sub-spaces in the spaces L p. Studia Math. 21, 161–176 (1962)
Komlós, J.: A generalisation of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967)
Levin, V.L.: Extremal problems with convex functionals that are lower semi-continuous with respect to convergence in measure. Dokl. Akad. Nauk SSSR 224, No 6, 1256–1259 (1975);
Levin, V.L.: Extremal problems with convex functionals that are lower semi-continuous with respect to convergence in measure. Englist transi.: Soviet math. Dokl. 16, No 5. 1384–1388 (1976)
Saadoune, M.: Une nouvelle extension en dimension infinie du Théorème de Komlös. Application: Compacité faible dans L1 x, convergence en mesure, Preprint, Université Ibnou Zohr, Agadir, Morocco 1995
Saadoune, M.: Compacité, Convergences et Approximations, Thèse de Doctorat d’Etat, Faculté des Sciences de Rabat, Juin 1996
Slaby, M.: Strong convergence of vector-valued pramarts and subpramarts, Probability and Math. Stat. 5, 187–196 (1985)
Talagrand, M.: Weak Cauchy sequences in L l (E). Amer. J. Math. 106, 703–724 (1984)
Ülger, A.: Weak compactness in L l (μ,X). Proc. Amer. Math. Soc. 103, 143–149 (1991)
Valadier, M.: Convergence en mesure et optimisation, Travaux du Séminaire d’Analyse convexe, Univ. Montpellier II (1976), exp 14.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this chapter
Cite this chapter
Castaing, C., Guessous, M. (1999). Convergences in L 1X (μ) . In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-65895-5_3
Download citation
DOI: https://doi.org/10.1007/978-4-431-65895-5_3
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-65897-9
Online ISBN: 978-4-431-65895-5
eBook Packages: Springer Book Archive