Convergence theorems for Pettis integrable functions and regular methods of summability

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:359

Article

Chakraborty, N. D.¹  Choudhury, Tanusree¹ 

[1] Univ Burdwan, Dept Math, Burdwan 713104, W Bengal, India

关键词: Pettis integrable functions;    Pettis integrable multifunctions;    Uniform integrability;    Theorem of Komlos;    Weak topology in P-1(mu, X);   

DOI  :  10.1016/j.jmaa.2009.05.014

来源: Elsevier

【 摘 要 】

We characterize the relatively sequentially compact subsets of P-1(mu, X), the space of all X-valued Pettis integrable functions, where X is a separable Banach space, for the weak topology of P-1(mu, 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 Komlos in P-1(mu, X), which is a generalization of a result of E.J. Balder in [E.J. Balder, Infinite-dimensional extension of a theorem of Komlos, 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(mu, X) analogous to the results of A. Amrani [A. Amrani, Lemme de Fatou pour l'integrale 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(mu, X) which are generalizations of some results of N.C. Yannelis [N.C. Yannelis, Weak sequential convergence in L-p(mu, X), J. Math. Anal. Appl. 141 (1989) 72-83] and A. Ulger [A. Ulger, Weak compactness in L-1(mu, X), Proc. Amer. Math. Soc. 113 (1991) 143-149]. (C) 2009 Published by Elsevier Inc.

【 授权许可】

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