GENERAL VECTOR MEASURE THEORY 7
of 2, then 21nF(En) is an unconditionally convergent series (with norm limit
F({JnEn)) in X. This property is shared by many noncountably additive vector
measures. For instance, if F: 2 -+ R is a nonnegative finitely additive measure,
then for any sequence (En) of pairwise disjoint members of 2 we have
t F(En) =
F(\J
En)^ F(Q),
so J^„F(En) oo. On the other hand, not all bounded vector measures have this
property; in fact, if 2 is infinite then the map F: 2 - B(2) given by F(A) = XA is
a bounded vector measure that lacks the above property. Because of its importance
in the theory of vector measures this property will be isolated.
DEFINITION
14. Let & be a field of subsets of the set Q and let F\& -* X be a
vector measure. F is said to be strongly additive whenever given a sequence (En)
of pairwise disjoint members of J^, the series J^^LiF(E„) converges in norm.
A family {FT: 2F - X \ z e T) of strongly additive vector measures is said to be
uniformly strongly additive whenever for any sequence (En) of pairwise disjoint
members of SF, then limj Ew=wFr(£m)|| = 0 uniformly in T e T.
Of course, countably additive vector measures on sigma-fields are strongly additive.
It is important to realize that in the definition of strong additivity the convergence
of the series 2^=iF(En) is unconditional in norm (since every subseries also con-
verges). It should also be noted that for families of countably additive measures on
a cr-field the concept of uniform strong additivity is precisely the familiar concept
of uniform countable additivity.
A wide but by no means exhaustive class of strongly additive vector measures is
furnished by
PROPOSITION
15. If F: 3F -* X is a vector measure of bounded variation, then F is
strongly additive.
PROOF.
If (En) is a sequence of pairwise disjoint members of J^, then
k / k
£||F(£„)|| ^ \F\{[j E„) ^ \F\(Q).
Thus £»ll^(iv)ll ^ \F\(Q) oo, andSn=1F(£n)is an absolutely convergent, hence
convergent, series in the Banach space X.
EXAMPLE
16. A countably additive, hence strongly additive, vector measure on a
sigma-field that is of unbounded variation over every nontrivial set. Let Q = [0, 1],
2 = Lebesgue measurable subsets of [0, 1], A = Lebesgue measure, 1 p oo,
and X = Lp[0, 1]. Define F : 2- Lp[0, 1] by F(E) = XE- Then it is easily checked
that if (En) is any sequence of pairwise disjoint Lebesgue measurable subsets of
[0, 1], then
U
En
- S F(EH)
\t
= A U
£ - ) - 0
as m -• oo. Thus Fis countably additive on the cr-field 2.
We now claim that if E c [0, 1] is Lebesgue measurable and X(E) 0, then
\F\(E) = oo. To prove this fix a positive integer n and pick disjoint measurable
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