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Title:  The Pettis Integral 
Author(s):  Geitz, Robert Frederick 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  The Pettis integral of a weakly measurable vectorvalued 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 vectorvalued function f : (OMEGA) (>) X in terms of the associated family {x*f : (VBAR)(VBAR) x* (VBAR)(VBAR) (LESSTHEQ) 1} of scalarvalued functions. This gives new insight into the various types of measurability for vectorvalued 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. 
Issue Date:  1980 
Type:  Text 
Language:  English 
Description:  85 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1980. 
URI:  http://hdl.handle.net/2142/68171 
Other Identifier(s):  (UMI)AAI8026497 
Date Available in IDEALS:  20141214 
Date Deposited:  1980 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois