This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/68171
Description
Title
The Pettis Integral
Author(s)
Geitz, Robert Frederick
Issue Date
1980
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
The Pettis integral of a weakly measurable vector-valued function is the most natural integral for use in Banach spaces. Although first defined over forty years ago, the integral has stubbornly defied analysis and has long been considered unmanageable. My thesis presents the first successful analysis of the Pettis integral. I show that a slight restriction on the measure spaces under consideration leads to a theory of Pettis integration very analogous to the theory of the better known, but more restrictive, Bochner integral. The resulting characterization of the Pettis integrable functions is much simpler than was previously believed possible.
The thesis falls naturally into three parts. I first consider a vector-valued function f : (OMEGA) (--->) X in terms of the associated family {x*f : (VBAR)(VBAR) x* (VBAR)(VBAR) (LESSTHEQ) 1} of scalar-valued functions. This gives new insight into the various types of measurability for vector-valued functions. I next make an extensive study of the properties of a function that are determined by the geometry of its range. Here I characterize the functions that are equivalent to strongly measurable functions and give the first necessary and sufficient conditions for a function to be Pettis integrable. The deep connection between perfect measure spaces and the Pettis integral also becomes apparent here. The final chapter of the thesis contains its most important results. Here I prove a dominated convergence theorem for the Pettis integral and characterize the Pettis integrable functions as limits, in a certain sense, of sequences of simple funtions.
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
