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