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Title:Nonstandard vector integrals and vector measures
Author(s):Zimmer, G. Beate
Doctoral Committee Chair(s):Loeb, Peter A.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:We describe an extension of the Bochner integral. Bochner integrable functions can be approximated by simple functions. Using Nonstandard Analysis, we investigate internal simple functions from an internal measure space to the nonstandard extension of a Banach space. We take suitable equivalence classes and identify the subspace of S-integrable functions with a space of functions from a Loeb space into the nonstandard hull of a Banach space. This space includes the Bochner integrable functions; it also includes nonmeasurable functions. For functions in this space we obtain an integral which generalizes the Bochner integral. For Banach lattices our integral coincides with an extension of the Bochner integral developed by Loeb and Osswald. We investigate the properties of the extended integral and characterize the space of extended integrable functions. The applications of this extended integral are mainly concerned with vector measures. One application is a generalized Radon-Nikodym derivative for all absolutely continuous vector measures of bounded variation.
Issue Date:1994
Rights Information:Copyright 1994 Zimmer, G. Beate
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9512611
OCLC Identifier:(UMI)AAI9512611

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