**Mathematical Surveys and Monographs**

Volume: 15;
1977;
322 pp;
Softcover

MSC: Primary 28;

**Print ISBN: 978-0-8218-1515-1
Product Code: SURV/15**

List Price: $70.00

AMS Member Price: $56.00

MAA Member Price: $63.00

**Electronic ISBN: 978-1-4704-1242-5
Product Code: SURV/15.E**

List Price: $66.00

AMS Member Price: $52.80

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# Vector Measures

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*J. Diestel; J. J. Uhl, Jr.*

In this survey the authors endeavor to give a comprehensive
examination of the theory of measures having values in Banach spaces. The
interplay between topological and geometric properties of Banach spaces and the
properties of measures having values in Banach spaces is the unifying
theme.

The first chapter deals with countably additive vector measures finitely
additive vector measures, the Orlicz-Pettis theorem and its relatives.
Chapter II concentrates on measurable vector valued functions and the
Bôchner integral.

Chapter III begins the study of the interplay among the Radon-Nikodým
theorem for vector measures, operators on \(L_1\) and topological
properties of Banach spaces. A variety of applications is given in the next
chapter.

Chapter V deals with martingales of Bôchner integrable functions and
their relation to dentable subsets of Banach spaces. Chapter VI is devoted to a
measure-theoretic study of weakly compact absolutely summing and nuclear
operators on spaces of continuous functions.

In Chapter VII a detailed study of the geometry of Banach spaces with the
Radon-Nikodým property is given. The next chapter deals with the use of
Radon-Nikodým theorems in the study of tensor products of Banach spaces.
The last chapter concludes the survey with a discussion of the Liapounoff
convexity theorem and other geometric properties of the range of a vector
measure.

Accompanying each chapter is an extensive survey of the literature and open
problems.

#### Table of Contents

# Table of Contents

## Vector Measures

- Contents vii8
- Foreword v6 free
- Introduction ix10 free
- I. General vector measure theory 116 free
- 1. Elementary properties of vector measures 116
- 2. Countably additive vector measures 1025
- 3. The Nikodým Boundedness Theorem 1429
- 4. Rosenthal's lemma and the structure of a vector measure 1833
- 5. The Carathéodory-Hahn-Kluvanek Extension Theorem and strongly additive vector measures 2540
- 6. Notes and remarks 3146

- II. Integration 4156
- III. Analytic Radon-Nikodým theorems and operators on L[sub(1)](μ) 5974
- IV. Applications of analytic Radon-Nikodým theorems 97112
- V. Martingales 121136
- VI. Operators on spaces of continuous functions 147162
- VII. Geometric aspects of the Radon-Nikodým property 187202
- 1. The Krein-Mil'man theorem and the Radon-Nikodým property 187202
- 2. Separable dual spaces, the Krein-Mil'man property and the Radon-Nikodým property 191206
- 3. Strongly exposed points and the Radon-Nikodým property 199214
- 4. The Radon-Nikodým property and the existence of extreme points for nonconvex closed bounded sets 203218
- 5. Notes and remarks 208223
- 6. Summary of equivalent formulations of the Radon-Nikodým property 217232
- 7. The Radon-Nikodým property for specific spaces 218233

- VIII. Tensor products of Banach spaces 221236
- IX. The range of a vector measure 261276
- Bibliography 277292
- Subject index 311326
- Author index 319334