Nonstandard vector integrals and vector measures

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Zimmer, G. Beate

Contributors dc:contributor

• Loeb, Peter A.

• Mathematics

Statement dc:rights

• Copyright 1994 Zimmer, G. Beate

Language dc:language

eng

Identifier

AAI9512611
(UMI)AAI9512611

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/19304