J. DIESTEL AND J. J. UHL, JR.
Since this holds for any partition %, the inequality | F\(\JnEH) ^ £ | F\(E„) obtains.
But now recall that | F\ is finitely additive and monotone on F. Thus for each n
±\F\{EM) = \F\(\jEk)^\F\({jEn).
This proves the reverse inequality Ln|^|(^n) = |^|(U»^»)
anc* s n o w s t n a t
\E\ is
countably additive on F.
COROLLARY
10. Let 2 be a o-field generated by a subfield &'. If F: 2 - X is a
countably additive vector measure of bounded variation and F\p is the restriction of
F to F then for each E e
J5",
one has
\F\r\(E) = \F\(E);
i.e., |F\ is the Caratheodory-Hahn extension of\F\?\ to 2.
PROOF.
Let p, be the countably additive Caratheodory-Hahn extension of \F\&\
to 2. Then for each £ e « f and for each x* e X* with ||JC* || ^ 1, one has
\x*F\,\(E) ^ (jiE).
But for the same x* and E,
\x*F\r\(E) = \x*F\(E).
Consequently one has |x*F\(E)£ p(E) for all Ee& and allx*eX* with
||JC*||
g 1.
It follows now from the facts that both \x*F\ and p are countably additive on 2
and the fact that 3F generates 2 that the inequality
\x*F\(E) ^ p(E)
holds for all E e 2 and all x* e X* with ||x* || g 1. But then, as was remarked be-
fore Proposition 9, the inequality |F|(£) g p(E) holds for all Ee 2.
On the other hand, it is plain that for any Ee& one has
\F\r\(E)£\F\(E).
Hence p{E) ^ |F|(£) for all Ee & and hence, since & generates 2, for all £ e l
Consequently p = \F\ and the proof is complete.
The next proposition presents two basic facts about the semivariation of a vector
measure.
PROPOSITION
11. Let F.3F - Xbea vector measure. Then for Ee&, one has
(a) ||F||(£) = sup|^^
w
)|}
where the supremum is taken over all partitions % of E into finitely many disjoint
members of & and all finite collections {en} satisfying \ek\ ^ 1; and
sup{||F(7/)||:E 2 He*} g ||F|(£)
K
' ^ 4sup{||F(#)||: E 2 HeP).
Consequently a vector measure is of bounded semivariation on 0 if and only if its
range is bounded in X.
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