GENERAL VECTOR MEASURE THEORY
PROOF.
If % = {Eh •••, Em} is a partition of E into pairwise disjoint members
of J* and eh •••, em are scalars such that \e\\, •••, |em| ^ 1 then
= sup{|^(S£„F(£„))|: x* e X*, ||x*|| £ l}
{2m
|
£ B
^ F ( £
B
) | : X * 6 X * , | | X * | | ^ 1
^ sup{|; |**F(£„)|: x* e X*, \\x* || g l}
£-||*1|(K
For the reverse inequality, let
JC*
e X* with
||JC*||
g 1 and suppose % = {£!,
£m} is a partition ofEe^ into pairwise disjoint members of J^. Then
2 \x*F(En)\ = £ (sgn x*F(En))x*F(En)
(f^sgnx*^,,))^
m I
Z (sgn x*F(En))F(En)\
This proves (a).
To prove (b) note that for Ee &, one has
sup{|F(//)|:£ 2 Hep) = sup{sup{|x»F(//)| :x*eJ!f*,||;t*|| ^ l ) : £ a / / ^ }
^ I1(£)-
Also, if 7 r = {£i, •••, Em} is a partition of a member E of ^ into pairwise disjoint
members of J5" and if x* e Z* satisfies ||x*|| g 1, then (in case X is a real Banach
space)
2 |**F(£M)| = 2 x*F(£n) - S x*F(En),
En^n »S7r+ »ejr~
where
x+
= {«: 1 ^ « ^ w, Jc*F(£n) ^ 0} and ^~ = {«: 1 ^ « ^ w, x*F(En) 0},
^ 2sup{||F(//)||:£ , 2 He&}
as required. In case A " is a complex Banach space, it is easy to see that a similar
estimate holds if the number 2 is replaced by the number 4. Simply split x*F into
real and imaginary parts and apply the real case.
In view of Proposition 11(b) a vector measure of bounded semivariation will
also be called a bounded vector measure.
As will be seen presently, it is easy to define the integral of a bounded measurable
function with respect to a bounded vector measure. To this end, let SF be a field of
subsets of 0 and F: & X be a bounded vector measure. If/ is a scalar-valued
simple function on Q9 s a y / = 2ji=iffiJte,- where at are nonzero scalars and Eh •••,
En are pairwise disjoint members of
J5",
define 7(/) = S?=iafF(£,). It is dreadfully
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