Abstract

We examine and extend several results about the universal Pettis integral property using a uniform approach via Edgar's ordering structure on Banach spaces. This reduces the problem from one concerning integrable functions to a purely Banach space setting.

Introduction

Originally defined in 1938 by B. J. Pettis, the Pettis integral lay dormant for forty years, elusive and seemingly banished to the realm of mathematical curiosities. Since 1978, however, substantial progress has been made, particularly for functions taking values in the dual of a Banach space. There are essentially two basic ways in which to study the integrability properties of functions. One is to concentrate on a particular function f : Ω → X and find conditions on f for which it will be Pettis integrable. This can be done, for example, in terms of an appropriate convergence of simple functions, looking at the core of the function [9], or examining the set {x*f : ∥x*∥ ≤ 1}. This approach has successfully lead to various characterizations of the Pettis integral, for which the monograph [16] by M. Talagrand provides an excellent reference. Recent results have also been attained by Riddle-Saab [12] and Andrews [1] on functions that are universally Pettis integrable, that is, functions defined on a compact Hausdorff space and Pettis integrable with respect to all Radon measures on that space.

The other approach is to study the Banach space X and find conditions on X for which all functions into X will be Pettis integrable under certain suitable measurability conditions.

Abstract

Let K be a line-closed measure-convex bounded subset of a Banach space so that every relatively closed separable subset of K is analytic. By constructing certain martingales it is proved that if K has the Radon-Nikodým property, then for every k0 in K there is a separable relatively closed convex subset K0 of K and a Borel probability measure μ supported on the extreme points of L, for every relatively closed separable convex subset L of K with L ⊃ K0, so that k0 is the barycenter of μ; μ is uniquely determined by k0 if and only if K is a simplex.

Introduction.

We give here a self-contained exposition of various known integral representation results, via the general theorem stated in the abstract. Both the existence and uniqueness parts of the theorem are obtained through the construction of certain martingales. Our main arguments and formulations are thus probabilistic in nature.

For the sake of orientation, we first recall the following representation result, due to E.G.F. Thomas [26].

Theorem.Let K be a closed bounded measure-convex Souslin subset of a locally convex space and assume that K has the Radon-Nikodým property (the RNP). Then for every x in K, there is a Borel probability measure μ supported on the extreme points of K so that x is the barycenter of μ; this μ is uniquely determined (for every given x) if and only if K is a simplex.

The Special Year in Modern Analysis at the University of Illinois was devoted to the synthesis and expansion of modern and classical analysis. The program brought together analysts from around the globe for intensive lectures and discussions, including an International Conference on Modern Analysis, held March 16–19, 1987. The Special Year's success is a tribute to the outstanding merits and professional dedication of the participants. Contributions to these Proceedings of the Special Year were solicited from the participants in order to record and disseminate the fruits of their activities. The editors are grateful to the contributors for their response, which accurately reflects the quality and substance of the Special Year. In keeping with the wide scope of topics treated, the contents of these Proceedings fell naturally into two interrelated volumes, covering “Analysis in Abstract Spaces” and “Analysis in Function Spaces”.

Thanks are due to the National Science Foundation, the Argonne Universities Association Trust Fund, the University of Illinois Campus Research Board, the University of Illinois Miller Endowment Fund, the University of Illinois Department of Mathematics, and J. Bourgain's Chair in Mathematics at the University of Illinois, without whose financial support the Special Year could not have taken place. Special thanks are also due to Professor Bela Bollobas, Consulting Editor at Cambridge University Press, and Mr. David Tranah, Senior Editor in Mathematical Sciences at Cambridge University Press, for the guidance and encouragement which made these Proceedings possible.

Introduction. In light of Enflo's famous counterexample to the approximation problem [3], the study of “weaker structures” has gained added importance. The most fruitful of these has been the bounded approximation property (B.A.P.) (see section 2 for the definitions), the πγ-property, and the finite dimensional decomposition property (F.D.D.P.). Johnson, Rosenthal and Zippin [11] examined certain relationships between these weaker structures. Since 1970, this paper has been the standard reference for people working in the area. Essentially no further positive progress has been made on the important problem of finding general conditions which imply the F.D.D.P. for a Banach space X.

Enflo's example [3] was the first in a long series of important counterexamples in the area. Figiel and Johnson [4] then showed the existence of a Banach space which has the approximation property (A.P.) but fails the B.A.P.. Lindenstrauss (see [14]) found a Banach space X with a basis so that X* is separable and fails the A.P. Recently, S. J. Szarek [18] constructed a Banach space with the F.D.D.P. which fails to have a basis.

In the sequel, we will see that the much ignored concept of commuting B.A.P. (C.B.A.P.) plays a central role in passing from “weaker structures” to a F.D.D. for a separable Banach space. In section 2 we give the definitions and review the work to date on these problems. Section 3 is a study of C.B.A.P. and what conditions imply its existence.