Adv. Math. Econ. 10, 1-29 (2007)

Komlós type convergence for random variables and random sets with applications to minimization problems

C. Castaing^{1} and M. Saadoune^{2}

1 Département de Mathématiques, Université Montpellier II, 34095 Montpellier Cedex 5, France

2 Département de Mathématiques, Université Ibnou Zohr, Lot. Addalha, B.P. 8106, Agadir, Maroc

Received: August 10, 2006 Revised: October 16, 2006

JEL classification: C61

Mathematical Subject Classification (2000): 49J40, 49J45, 46N10

Abstract. Let E be a separable super reflexive Banach space and let (Ω, F, μ) be a complete probability space. We state some Komlós type theorems in the space L_{cwk(E)}^{0}(Ω, F, μ) of E-valued random variables and a version of Komlós slice theorem in the space L_{cwk(E)}^{0}(Ω, F, μ) of convex weakly compact random sets. Weak Komlós type theorems for some unbounded sequences in L_{F}^{1} (Ω, F, μ) and L_{F′}^{1}[F](Ω, F, μ) 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 L_{E}^{0}(Ω, F, μ) are presented. Further Minimization problems and Min-Max type results involving saddle-points and Young measures are also investigated.

Key words: Biting Lemma, Komlós convergence, minimization, Min-Max, saddle points, young measures