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
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.
Previous Page Next Page