INTRODUCTION XI

The first chapter deals with countably additive and finitely additive vector meas-

ures. The basic behavior of countably additive measures is presented from the view-

point of the fundamental work of Bartle, Dunford and Schwartz and Kluvanek.

We base the theory of finitely additive vector measures on Rosenthal's lemma.

With the help of this lemma, we examine the roles of the spaces c0 and lx in the the-

ory of vector measures. Included here are the Vitali-Hahn-Saks-Nikodym theorem

and the Nikodym boundedness theorem for finitely additive vector measures.

The second chapter, which is for the most part independent of Chapter I, is de-

voted to measurable functions with values in Banach spaces and the problem of

integrating them. The Bochner integral receives most of the attention, but basic

material on the Pettis, Dunford and Gel'fand integrals is found in this chapter.

The Radon-Nikodym theorem for the Bochner integral is the subject of Chapter

III. We try to follow the genetic approach of treating the Radon-Nikodym theorem

and Lx operator theory as one unified theory. The analytic (i.e., topological) aspects

of the Radon-Nikodym theorem are found here. The roles of compact operators

on L

b

weakly compact operators on L\, reflexive spaces, separable dual spaces and

weakly compactly generated dual spaces in the Radon-Nikodym property are also

discussed here.

Chapter IV continues with a potpourri of applications of the Radon-Nikodym

theorem for the Bochner integral. The duals of the Z^-spaces of Bochner integrable

functions are derived and weak compactness in the space of Bochner integrable

functions is discussed. The relationship between differentiate vector-valued func-

tions of a real variable and the Radon-Nikodym theorem is next. Then the rela-

tionship between the classical integral operators on Lp and the Bochner integral is

surveyed. The chapter concludes with the Lewis Stegall theorem on complemented

subspaces of Lv

Martingales of Bochner integrable functions headline Chapter V. In addition to

martingale convergence theorems, we observe a basic phenomenon in the theory

of vector measures. Through a meld of the Radon-Nikodym theorem, elementary

martingale theory, and geometry of Banach spaces, we see the Radon-Nikodym

property transfer itself from an analytic property of Banach spaces to a geometric

property of Banach spaces. This is the link between geometry and measure theory

in Banach spaces.

Structural properties of operators on spaces of continuous functions C(0) are

under scrutiny in Chapter VI. The basic work of Bartle, Dunford and Schwartz

and Grothendieck is discussed from the viewpoint of Chapter I in the first part.

The second part deals with absolutely summing and nuclear operators on C(Q)

and their relationship with the Radon-Nikodym theorem. Included here is a dis-

cussion of Pietsch integral operators on Banach spaces.

The seventh chapter builds on the martingale theory of Chapter V to give an ex-

position of the repercussions of the Radon-Nikodym theorem in the geometry of

Banach spaces. Studied here are the relationships among the Radon-Nikodym theo-

rem, the Krein-Mirman theorem, properties of strongly exposed points, and other

extreme point phenomena. This chapter can be read directly after Chapter V.

Tensor products of Banach spaces and how the Radon-Nikodym theorem can be

used within the theory of tensor products to study Banach spaces is the theme of

Chapter VIII.