INTRODUCTION Xlll

ing this extra bit of generality is not worth the space and its inclusion would obscure

the exposition in a mass of trivial details.

Although we treat in detail the representation of operators on Ll5 L^ and C(Q),

the representation of the general operator on Lp for 1 p oo is conspicuously

absent. Our reason for this is that we do not know any applications of this repre-

sentation theory. However we do study some important classes of operators on Lp.

A third omission is the integration and differentiation theory for functions that

are not norm (strongly) measurable. We are quick to admit that an extensive theory

exists for such functions. We know of very few honest applications of this theory.

Additional omissions include measures with values in linear topological spaces

other than Banach spaces, orthogonally scattered measures, vector-valued stochas-

tic processes (other than martingales) and the lifting theory for vector-valued func-

tions. A very serious omission is most of the material found in the monograph of

Igor Kluvanek and Greg Knowles (Vector measures and control systems, North-

Holland, Amsterdam, 1976). Those who desire more material on the range of a

vector measure than found in Chapter IX or who want to study infinite dimensional

control theory should consult this spendid volume.

During the preparation of this survey, we have been helped immeasurably by a

number of our colleagues who have freely contributed their advice and criticism.

A partial list of those to whom we owe our heartfelt thanks is: R. G. Bartle, W. J.

Davis, M. M. Day, L. Dor, G. A. Edgar, B. T. Faires, T. Figiel, J. Hagler, R. E.

Huff, J. A. Johnson, W. B. Johnson, N. J. Kalton, R. Kaufman, I. Kluvanek, G.

Knowles, D. R. Lewis, H. B. Maynard, P. D. Morris, T. J. Morrison, R. E. Olson,

N. T. Peck, A. Pelczyriski, A. L. Peressini, B. J. Pettis, R. R. Phelps, H. P. Rosenthal,

E. Saab, C. J. Seifert, T. W. Starbird, F. E. Sullivan, J. B. Turett and A. Vento.

We also owe a measure of gratitude to the editors of this series, especially R. G.

Bartle, P. R. Halmos and M. Rosenlicht for wrestling with our sometimes uncon-

ventional style. We are much indebted to Carolyn Bloemker and Kathy Morrison

for typing this survey. Their job was not easy. Finally we thank Linda Diestel for

putting up with both of us.

As we progressed in the study of the history of the basic theorems of the theory

of vector measures, we were not surprised by learning that most of them, in one

way or another, have their origins in the fertile mind of one man, B. J. Pettis, who

was kind enough to give us the benefit of his wisdom on many matters and to agree

to write the foreword. To this mathematician and gentleman we dedicate our work.

KENT, OHIO J. DIESTEL

URBANA, ILLINOIS

J. J.

UHL, JR.