Chapter IX concludes the survey with a discussion of the Liapounofif convexity
theorem and other geometric properties of the range of a vector measure.
At the end of each chapter is a section called "Notes and remarks." These sec-
tions, which are modeled after similiar sections in Dunford and Schwartz, attempt
to discuss the original and subsequent versions of the results presented in the
chapter in question. Sometimes they contain additional results, often with proofs,
that could not be fitted into the main text. In each of these sections, there is an
attempt to discuss additional results that bear on the theorems presented in the text.
Sometimes these discussions contain terminology that is not defined in the text.
Usually the terminology is standard. When in doubt, the reader should consult the
appropriate reference.
We envision that this survey will be useful in a variety of ways. Those who want
to study the Radon-Nikodym theorem and its relation to the topological and
geometric structure of Banach spaces should read Chapters II, III, V and VII.
Those who have an additional interest in applications of the Radon-Nikodym
theorem may also want to look at Chapters IV, VI and VIII. Those who want to
study measures of unbounded variation can read Chapter I, thefirstpart of Chapter
VI and Chapter IX. We have attempted to minimize the introduction of weighty
terminology and notation. Thus it should be possible for someone who has not
read the early chapter to be able to understand the content of a theorem in a late
chapter with a minimum of frustration and page turning. We hope that this will
make this survey useful for spot references.
The numbering of theorems is the same as in Dunford and Schwartz; thus The-
orem V.2.6 is the sixth numbered item in the second section of the fifth chapter.
Within the second section of thefifthchapter this theorem is referred to as Theorem
6; within the other sections of the fifth chapter this theorem is referred to as
Theorem 2.6. Elsewhere it is referred to as V.2.6.
We hope our terminology is standard. To prevent any doubt let us fix some
terminology. When we say that (0, 2 fi) is a measure space, we mean that p. is
an extended real-valued nonnegative countably additive measure defined on a
(7-field 2 of subsets of a point set Q. The triple (0, 2, JLL) is called a finite measure
space if it is a measure space and fi(Q) is finite. A subset of a Banach space is called
relatively norm (weakly) compact if its norm (weak) closure is norm (weakly) com-
pact. A subset of a Banach space is called conditionally weakly compact if every
sequence in it has a weakly Cauchy subsequence. If X and Y are Banach spaces,
if (A", Y) stands for the space of bounded linear operators from X to Y; the space
X contains a copy of Y if X has a subspace that is linearly homeomorphic to Y.
There is one theorem that will be used from time to time and that may be unfamiliar
to some readers. This is Stone's representation theorem which says that if & is a
field of subsets of a point set Qy then there is a compact Hausdorff space fllt a field
2\ consisting of subsets of Qx that are both closed and open, and there is a Boolean
isomorphism between & and &x. The field &x will be called the Stone representa-
tion algebra of $F.
Some readers may find some serious omissions in this survey. We deal with finite
measure spaces only. Many of the theorems we present here have extensions to
more general situations; some do not. When a theorem has an extension to more
general situations, its extension is usually a routine extension. We feel that captur-
Previous Page Next Page