I. GENERAL VECTOR MEASURE THEORY

Grubby set-theoretic manipulations cannot be avoided in measure theory and

most of them are found in this chapter. This is not all bad because they are at the

base of a number of fundamental theorems of vector measure theory and Banach

space theory. The first section introduces the notions of variation, semivariation,

strong additivity (s-boundedness) and countable additivity. Also in this section is a

brief look at integration with respect to a vector measure. Most of this section

consists of straightforward manipulations of definitions.

§2 is a basic section which examines the essential properties of countably addi-

tive vector measures on ^--fields. Here the Bartle-Dunford-Schwartz theorems are

found. §3 continues with an exposition of the Nikodym Boundedness Theorem.

Rosenthal's lemma forms the core of §4 which is one of the major sections of

this book. In this section the interchange between the spaces c0 and /« and vector

measure theory begins to emerge. From this perspective, the Orlicz-Pettis theorem,

the Bessaga-Pelczynski c0 theorem, and the Vitali-Hahn-Saks-Nikodym theorem

are deduced very simply.

The last section begins with the Caratheodory-Hahn-Kluvanek Extension The-

orem for vector measures. Then by Stone space arguments it is shown that strongly

additive vector measures have almost all the properties of countably additive vector

measures except countable additivity. The section concludes with derivations of the

Yosida-Hewitt and Lebesgue decomposition theorems for vector measures.

1, Elementary properties of vector measures. This section deals with basic straight-

forward properties of vector measures. The familiar notions of variation and count-

able additivity are introduced together with the concepts of semivariation and

strong additivity. Finally, an elementary integral is introduced to help establish

a basic relationship between vector measures and operators on spaces of bounded

measurable functions.

DEFINITION

1. A function F from a field & of subsets of a set 0 to a Banach

space X is called a, finitely additive vector measure, or simply a vector measure, if

whenever Ex and E2 are disjoint members of J5" then F(EX U £2) = ^(£1) +

If, in addition, i^USLi En) = En=i F{En) in the norm topology of X for all

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http://dx.doi.org/10.1090/surv/015/01