6
J. DIESTEL AND J. J. UHL, JR.
boring to show that this formula defines a linear map TF from the space of simple
functions of the above form into X and we leave this as an exercise for masochists.
Moreover, if/is as above and /3 = sup{|/(co) |: co e £?}, then
!W)||=|i,^(£,)
\\t {caimE.)
g isMkfl)
by Proposition 11(a). Thus, if the space of simple functions over J*" is given the
supremum norm, TF acts on this space as a continuous linear operator with ||rF||
S II^IIOO). Another look at Proposition 11(a) and the above calculations shows that
in fact||rF|i = ||F||(0).
Next note that since TF is continuous and linear from the simple functions mod-
eled on 3F to X, TF has a unique continous linear extension, still denoted by TF,
to B(^), the space of all scalar-valued functions on 0 that are uniform limits of
simple functions modeled on F. (Note that in the case J^ is a ^-field, B(&) is
precisely the familiar space of bounded ^-measurable scalar-valued functions
defined on Q.)
This discussion allows us to make
DEFINITION
12. Let & be a field of subsets of the set Q and let F: SF - X be a
bounded vector measure. For each/e B(F), \f dF is defined by
j.
JdF = TF(f)
where TF is as above.
The general subject of integration with respect to a vector measure will be dis-
cussed later. For the present, this cheap integral as defined above has some conven-
ient properties and uses. It is, of course, linear in/(and also in F) and satisfies
j
fdF 1*11(0).
Moreover, if
x*eAr*,then
x* j / dF §fdx*F holds; indeed, for simple functions
/this equality is trivial and density of simple functions in B(^) proves the identity
for all fe B(?F). The following formality will allow various properties of vector
measures to be translated into properties of linear operators and vice versa.
THEOREM
13. Let ^(2) be a field (resp. o-field) of subsets of the set Q.
Suppose ju is an extended real-valued nonnegative finitely additive measure on 2.
Then there is a one-to-one linear correspondence between y?(B(3F)\ X) {resp.
£{LJ(fj)\ X)) and the space of all bounded vector measures F: fF -• X (resp. all
bounded vector measures F: 2- X that vanish on /u-null sets) defined by F- TF
ifTFf= j/dFfor allfeB(F) (resp. L ^ ) ) . Moreover \\TF\\ = ||F||(fl).
The proof is an easy combination of the observations and propositions preced-
ing the statement of the theorem and is left as an exercise.
One obvious property of a countably additive vector measure F defined on a
a-field ^(with values in X) is that if (En) is a sequence of pairwise disjoint members
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