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Softcover ISBN:  9780821815151 
Product Code:  SURV/15 
List Price:  $129.00 
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Book DetailsMathematical Surveys and MonographsVolume: 15; 1977; 322 ppMSC: Primary 28
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 OrliczPettis 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 RadonNikodým 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 measuretheoretic 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 RadonNikodým property is given. The next chapter deals with the use of RadonNikodým 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

Chapters

I. General vector measure theory

II. Integration

III. Analytic RadonNikodým theorems and operators on $L_1(\mu )$

IV. Applications of analytic RadonNikodým theorems

V. Martingales

VI. Operators on spaces of continuous functions

VII. Geometric aspects of the RadonNikodým property

VIII. Tensor products of Banach spaces

IX. The range of a vector measure


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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 OrliczPettis 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 RadonNikodým 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 measuretheoretic 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 RadonNikodým property is given. The next chapter deals with the use of RadonNikodým 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.

Chapters

I. General vector measure theory

II. Integration

III. Analytic RadonNikodým theorems and operators on $L_1(\mu )$

IV. Applications of analytic RadonNikodým theorems

V. Martingales

VI. Operators on spaces of continuous functions

VII. Geometric aspects of the RadonNikodým property

VIII. Tensor products of Banach spaces

IX. The range of a vector measure