GENERAL VECTOR MEASURE THEORY 3
sets of F each with positive measure such that \JnEn = E. Set %n {Eh F2, •,
En-h {Jg=»Ek}. Then for each n,
2 11^)11 = SWll l-lh^ Ek[Jk=nEk\\=n.
Accordingly |F|(F) is infinite. The fact that this measure is of bounded semivaria-
tion is a consequence of
EXAMPLE
7. Vector measures of bounded semivariation. Let T: LJQ, 1] X be a
continuous linear operator and for a Lebesgue measurable set EG [0, 1] define
F(E) = J(^£). If x* e * * and ||JC* || ^ 1 and ^ is a partition of [0, 1] into Lebesgue
measurable sets, then
2 \x*F(A)\ = 2 \x*TXA\ = 2 sgz
X*TXAX*TXA
A^TC A^K
A^it
= x*T(Z(sgnx*TXA)XA)
2 ( s g n x * 7 ^ ) ^
^ **r
m.
Thus Fis of bounded semivariation.
EXAMPLE
8. A measure of unbounded semivariation. Although little can be said of
such measures it is worth noting that a vector measure (in fact, a real-valued meas-
ure) need not be of bounded semivariation. Indeed, if J^ is the field of subsets of
iV, the positive integers, consisting of sets that are either finite or have finite com-
plements, then the measure F: 3F R defined by
F(E) = cardinality of F, if F is finite,
= cardinality of N\E, if N\E is finite,
produces an example of real-valued measure with unbounded semivariation.
Of some use is the easily verified fact that if F: SF - X is a vector measure of
bounded variation, then a nonnegative measure fz onFis the variation | F| of F if
and only if p satisfies: (i) | x*F\ (F) ^ ju(E) for all F e & and all x* e X* with ||JC* ||
^ 1, and (ii) if X: SF -+ R is any measure satisfying |x*F|(F) g A(F) for all Ee &
and all x* e X* with ||x*|| ^ 1 then fx(E) S A(F) for all Ee2. In terms of the
lattice structure of the space of set functions, | F\ is the least upper bound (if it
exists) of the collection {|x*F|: JC* e X* and ||x*|| ^ 1}.
PROPOSITION
9. A vector measure of bounded variation is count ably additive if and
only if its variation is also countably additive.
PROOF.
Suppose F: & -+ Zis of bounded variation. Since ||F(F)|| ^ |F|(F) for
each F e J5", it is plain that F is countably additive if | F\ is countably additive.
Conversely, suppose that F: J^ -• A' is a countably additive vector measure of
bounded variation. Let (Fn) be a sequence of pairwise disjoint members of SF such
that \JnEn G & and let % be a partition of (JnF„ into pairwise disjoint members of
IF. Then
2 l ^ ) | | = £ I I ^ n U^)|| = L \\ZF(A()EM)\\ iZL \\F(Af]En)\\
A^n A^n II \ » /II ,4ejr H w II A^it n
= Li:\\F(Af]En)\\ S2|F|(£„).
» A^n n
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