2
J. DIESTEL AND J. J. UHL, JR.
sequences (En) of pairwise disjoint members of & such that \J%Li En e J% then F
is termed a countably additive vector measure or simply, F is countably additive.
EXAMPLE 2. A finitely additive vector measure. Let T: £«,[(), 1] -+ Xbt a continu-
ous linear operator. For each Lebesgue measurable set E c [0, 1], define F(E) to
be T(XE) (XE denotes the characteristic or indicator function of E). Then by the
linearity of F, Fis seen to be a finitely additive vector measure which may—even
in the case that Xh the real numbers—fail to be countably additive. The simplest
such general example of a noncountably additive measure is provided by consider-
ing any Hahn-Banach extension to LJO, 1] of a point mass functional on C[0, 1].
EXAMPLE
3. A countably additive vector measure. Let T: Lj[0, 1] -+ A'be a continu-
ous linear operator. Again define F(E) = T(%E) for each Lebesgue measurable set
E c [0, 1]. Then F is evidently finitely additive. Moreover, for each F, one has
||F(F)|| g A(E)\\T\\. Consequently, if (£n)£Li is a sequence of disjoint Lebesgue
measurable subsets of [0, 1], then
(
oo \ m II I | / oo \
US.) " Lf(En) =
H m
W U
E»)
/ °°
limfl (J
gli m A En)T = 0.
lim
All the measures produced via Example 3 have a property isolated by the first
part of
DEFINITION 4. Let F: & -* X be a vector measure. The variation of F is the
extended nonnegative function | F\ whose value on a set E e & is given by
\F\(E) = s u p 2 \\F(A)l
where the supremum is taken over all partitions % of E into a finite number of
pairwise disjoint members of !F. If | F\(Q) oo, then F will be called a measure of
bounded variation.
The semivariation of F is the extended nonnegative function ||F|| whose value on
a set F e J^ is given by
||F||(F) = sup{|**F|(F):** e X*, \\x*\\ g 1},
where \x*F\ is the variation of the real-valued measure x*F. If ||F||(fi) oo, then
F will be called a measure of bounded semivariation.
Direct verifications show that the variation of F is a monotone finitely additive
function on g?9 while the semivariation of F is a monotone subadditive function on
F. Also it is easy to see that for each E e ZF one has ||F||(F) g |F|(F).
Now Examples 2 and 3 will be re-examined from the point of view of variations
and semivariations.
EXAMPLE
5. A measure of bounded variation. Let F be a measure of the type dis-
cussed in Example 3. Since ||F(F)|| g \T\l{E\ it is plain that \F\(E) g ||71|A(F),
so that Fis of bounded variation.
EXAMPLE
6. A measure of bounded semivariation but not of bounded variation.
Let 2 be the a-field of Lebesgue measurable subsets of [0, 1] and define F: 2 -
FJO, 1], by F(E) = XE• H" F e 2 and A(F) 0, select a disjoint sequence (£„) of sub-
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