Universal Pettis Integrability

Abstract

Since the invention of the Pettis integral over forty years ago [11], the problem of recognizing the Pettis integrability of a function against an individual measure has been much studied [5, 6, 7, 8, 9, 20]. More recently, Riddle-Saab-Uhl [14] and Riddle-Saab [13] have considered the problem of when a function is integrable against every Radon measure on a fixed compact Hausdorff space. These papers give various sufficient conditions on a function that ensure this universal Pettis integrability. In this paper, we see how far these various conditions go toward characterizing universal Pettis integrability. We base our work on a w*-analogue of the core of a vector-valued function [8].We also give some sufficient conditions that ensure that a Banach space has the so-called universal Pettis integral property (UPIP) and consider some particular examples of spaces with this property. It is interesting that in these examples some of the special set theoretic axioms that play an important role in the study of the stronger Pettis integral property [6, 7] make an appearance.