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
\\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
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
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
fdF 1*11(0).
Moreover, if
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.
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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