X INTRODUCTION
little more than formal generalizations of the scalar theory. Representation theory
for operators on function spaces became the vogue. But all too often these repre-
sentation theories gave no new information about the operators they represented.
During the fifties and early sixties the theory of vector measures languished in
sterility.
There were one or two bright spots. In the mid-fifties, A. Grothendieck used
the then ignored vector measure theory of the late thirties and early forties to launch
a monumental study of linear operators. The repercussions of Grothendieck's work
are still being felt today. Also in the mid-fifties, Grothendieck and (independently)
R. G. Bartle, N. Dunford and J. T. Schwartz studied operators on spaces of con-
tinuous functions and proved the first important theorems in the theory of vector
measures in some fifteen years. But the unfortunate truth is that, aside from I.
Kluvanek and a few others, no one followed their lead.
In the early sixties, largely through the pioneering work of A. Pelczyriski and J.
Lindenstrauss, Banach space theory came back to life and today has re-emerged as
a deep and vigorous area of mathematical inquiry. Vector measure theory did not
come around so quickly.
In the mid-sixties, N. Dinculeanu gave an intensive study of many of the the-
orems of vector measure theory that had been proven between 1950 and 1965.
Dinculeanu's monograph was the catalytic agent that the theory of vector measures
needed. Upon the appearance of Dinculeanu's book, interest in vector measures
began to grow. It was not long before a number of mathematicians addressed them-
selves to the basic unsolved problems of vector measure theory. The study of the
Radon-Nikodym theorem for the Bochner integral and the Orlicz-Pettis theorem
served to re-establish the links between vector measures and the analytic, geometric
and isomorphic theory of Banach spaces. Today the theory of vector measures
stands as a hearty cousin and proud servant of the theory of Banach spaces. This
survey is a report on how this has come about.
We endeavor to give a comprehensive survey of the theory of vector measures
as we see it. It is our overriding desire to emphasize the fruitful (and we think
exciting) interplay between properties of Banach spaces and measures taking
values in Banach spaces. Thus the exposition of the relationships among vector
measures, operators on L\, operators on spaces of bounded measurable functions,
topological structure of Banach spaces and geometric structure of Banach spaces
is our unifying theme. We feel that any attempt to divorce vector measures from
these latter areas would wallow in artificiality.
This survey is written for the student as well as the advanced mathematician.
Much of it originated in lectures given by the authors at Kent State University and
the University of Illinois. Other parts of the survey have grown from conversa-
tions with our colleagues in the classroom and other places where mathematicians
gravitate to talk. We assume that the reader has some familiarity with basic Banach
space theory as presented in Chapters II, V and VI of Linear operators by Dunford
and
Schwartz1
and with basic measure theory as presented in Battle's Elements of
integration or Halmos's Measure theory. Other than this, this survey is self-
contained.
'It may be noted that much of this survey is an outgrowth of Chapters IV and VI of Dunford
and Schwartz.
Previous Page Next Page