Adv. Math. Econ. 11, 11-44 (2008)

Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions

Charles Castaing^{1}, Christian Hess^{2} and Mohamed Saadoune^{3}

1 Département de Mathématiques, Université Montpellier II, Place E. Bataillon, 34095 Montpellier cedex, France

2 Centre de Recherche Stratégies et Dynamiques Financières, Université Paris Dauphine, 75775 Paris cedex 16, France

3 Département de Mathématiques, Université Ibn Zohr, Lot. Addakhla, B.P. 8106, Agadir, Maroc

Received: September 11, 2006 Revised: May 30, 2007

JEL classification: C62, C65

Mathematical Subject Classification (2000): 28B20, 26E25, 46E30, 46A50, 52A07, 54C60, 54C65, 91B50

Abstract. Various notions of tightness for measurable multifunctions are introduced and compared. They are used to derive results on the existence of integrable selections for the sequential weak upper limit of a sequence of multifunctions. Similar questions are examined for multifunctions with values in a dual space. Some results are particularized in the single-valued case, and applications to the multidimensional Fatou Lemma, both in the primal and in the dual space, are derived. This is achieved under conditions weaker than or non comparable to L^{1}-boundedness.

Key words: Tightness conditions, upper limit of a sequence of multifunctions, measurable selection, integrable selection,Fatou’s Lemma in several dimensions