Convergence theorems for Pettis integrable functions and regular methods of summability

We characterize the relatively sequentially compact subsets of P1(μ,X), the space of all X-valued Pettis integrable functions, where X is a separable Banach space, for the weak topology of P1(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 P1(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 P1(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 P1(μ,X) which are generalizations of some results of N.C. Yannelis [N.C. Yannelis, Weak sequential convergence in Lp(μ,X), J. Math. Anal. Appl. 141 (1989) 72–83] and A. Ülger [A. Ülger, Weak compactness in L1(μ,X), Proc. Amer. Math. Soc. 113 (1991) 143–149].