Abstract
Let E be a separable super reflexive Banach space and let \( \left( {\Omega ,\mathcal{F}{\text{,}}\mu } \right) \) be a complete probability space. We state some Komlós type theorems in the space \( \mathcal{L}_E^{\text{0}} \left( {\Omega ,\mathcal{F}{\text{,}}\mu } \right) \) of E-valued random variables and a version of Komlós slice theorem in the space \( \mathcal{L}_{cwk\left( E \right)}^{\text{0}} \left( {\Omega ,\mathcal{F},\mu } \right) \) of convex weakly compact random sets. Weak Komlós type theorems for some unbounded sequences in \( \mathcal{L}_F^{\text{1}} \left( {\Omega ,\mathcal{F},\mu } \right){\mathbf{ }}{\text{and}}{\mathbf{ }}\mathcal{L}_F^{\text{1}} \left[ F \right]\left( {\Omega ,\mathcal{F},\mu } \right) \) when F is a separable Banach space are also stated. A Fatou type lemma in Mathematical Economics and minimization problems on convex and closed in measure subsets of \( \mathcal{L}_E^{\text{0}} \left( {\Omega ,\mathcal{F},\mu } \right) \) are presented. Further Minimization problems and Min-Max type results involving saddle-points and Young measures are also investigated.
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Amrani, A., Castaing, C., Valadier, M.: Méthodes 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.: New fundamentals of Young measure convergence. In: Calculus of variations and optimal control, Haifa (1998), Chapman Hall, Boca Raton 24–48 (2000)
Balder, E.J.: A general approach to lower semicontinuity result and lower closure in optimal control theory. SIAM J. Control Optim. 22, 570–598 (1984)
Balder, E.J., Hess, Ch.: Two generalizations of Komlós theorem with lower closuretype aplications. J. Convex Anal. 3, 25–44 (1996)
Benabdellah, H., Castaing, C.: Weak compactness and convergences in \( L_E^1 ,\left[ E \right] \) . Adv. Math. Econ. 3, 1–44 (2001)
Beer, G.: Topologies on closed and closed convex sets. Kluwer, Dordrecht 1993
Bourgain, J.: The Komlós theorem for vector valued functions. Wrije Universiteit, Brussels, 1979/12
Grothendieck, A.: Espaces vectoriels topologiques. Publicacão da Sociedade de Matemática de Sao Paulo (1964)
Castaing, C.: Quelques résultats de convergence des suites adaptées. Sém. Anal. Convexe Montp. 17, 1–24 (1989)
Castaing, C.: Topologie de la convergence uniforme sur les parties uniformément inteǵrables de \( L_E^1 \) et théorèmes de compacité faible das certains espaces du type Kothe-Orlicz. Séminaire d’ Analyse Convexe 10, 1–27 (1980)
Castaing, C.: Weak compactness and convergences in Bochner and Pettis integration. Vietnam J. Math. 24(3), 241–286 (1996)
Castaing, C., Ezzaki, F.: Convergences for convex weakly compact random sets in B-convex reflexive Banach spaces. Atti Sem. Mat. Fis. University of Modena, XLVI, 123–149 (1998)
Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces. With applications in control theory and probability theory. Kluwer Dordrecht 2004
Castaing, C., Guessous, M.: Convergences in \( L_X^1 \left( \mu \right) \) . Adv. Math. Econ. 1, 17–37 (1999)
Castaing, C., Hess, Ch., Saadoune, M.: Tightness conditions and Integrability of the sequential weak upper limit of a sequence of Multifunctions. Working paper 2005
Castaing, C., Hess, Ch., Saadoune, M.: On various versions of Fatou lemma. Working paper 2006
Castaing, C., Saadoune, M.: Dunford-Pettis-types theorem and convergences in set-valued integration, J. Nonlinear Convex Anal. 1(1), 37–71 (2000)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Springer, Berlin, 580, 1977
Garling, D.J.H.: Subsequence principles for vector-valued random variables. Math. Proc. Cambridge Philos. Soc. 86, 301–311 (1979)
Komlós, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar 18, 217–229 (1967)
Krupa, G.: Komlós theorem for unbounded random sets. Set-Valued Anal. 8(3), 237–251 (2000)
Moreau, J.J.: Théorèmes “inf-sup”. C. R. Acad. Sci. Paris, 258, 2720–2722 (1964)
Saadoune, M.: A new extension of Komlós theorem in infinite dimensions. Application: weak compactness in \( L_X^1 \) . Portugaliae Math. 55, 113–128 (1998)
Sion, M.: On general minimax theorems, Pacific J. Math. 8, 171–176 (1958)
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Castaing, C., Saadoune, M. (2007). Komlós type convergence for random variables and random sets with applications to minimization problems. In: Kusuoka, S., Yamazaki, A. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 10. Springer, Tokyo. https://doi.org/10.1007/978-4-431-72761-3_1
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