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Title:The Pettis Integral
Author(s):Geitz, Robert Frederick
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:1980
Description:85 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.
Other Identifier(s):(UMI)AAI8026497
Date Available in IDEALS:2014-12-14
Date Deposited:1980

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