Convergence theorems for Pettis integrable functions and regular methods of summability

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• 作者：N.D. Chakraborty ; Tanusree Choudhury
• 关键词：Pettis integrable functions ; Pettis integrable multifunctions ; Uniform integrability ; Theorem of Koml&#xf3 ; s ; Weak topology in ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4W8TWBY-6
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2009
• 出版时间：1 November 2009
• 年：2009
• 卷：359
• 期：1
• 页码：95-105
• 全文大小：207 K

文摘

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 L_p(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].