Convergence theorems for Pettis integrable functions and regular methods of summability

Abstract

We characterize the relatively sequentially compact subsets of P 1 ( μ , X ) , the space of all X -valued Pettis integrable functions, where X is a separable Banach space, for the weak topology of P 1 ( μ , X ) by using the regular methods of summability. These characterizations are alternative descriptions of the results already done by Amrani and Castaing in [A. Amrani, C. Castaing, Weak compactness in Pettis integration, Bull. Pol. Acad. Sci. Math. 45 (2) (1997) 139–150]. We also study the theorem of Komlós in P 1 ( μ , X ) , which is a generalization of a result of E.J. Balder in [E.J. Balder, Infinite-dimensional extension of a theorem of Komlós, Probab. Theory Related Fields 81 (1989) 185–188, Theorem B]. We also prove some convergence theorems by applying the theorem. We also prove convergence theorems in P 1 ( μ , X ) analogous to the results of A. Amrani [A. Amrani, Lemme de Fatou pour l'intégrale de Pettis, Publ. Math. 42 (1998) 67–79] and H. Ziat [H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Pol. Acad. Sci. Math. 45 (2) (1997) 123–137]. Finally, we prove some convergence theorems in P 1 ( μ , X ) which are generalizations of some results of N.C. Yannelis [N.C. Yannelis, Weak sequential convergence in L p ( μ , X ) , J. Math. Anal. Appl. 141 (1989) 72–83] and A. Ülger [A. Ülger, Weak compactness in L 1 ( μ , X ) , Proc. Amer. Math. Soc. 113 (1991) 143–149].