A N N A L E S SO C IE T A T IS M A TH EM A TIC A E P O LO N A E Series I : C O M M EN TA TIO N ES M A TH EM A T IC A E X I X (1977) R O C Z N IK I P O L S K IE G O TO W A RZY STW A M ATEM ATYCZNEGO
Séria J : P R A C E M A TEM A TY CZN E X I X (1977)
E . U
rbanski(Poznan)
Weak compactness oî range oî a set-valued measure
In [2], E. G. Bartle, Ж. Dunford and J. T. Schwartz showed that if E is a Banach space and E is a cr-algebra, then the range of a vector
valued measure m: E-^-E is o(E , _Z2')-relatively compact. This was gen
eralized in [6] by I. Tweddle, who proved that the result remains valid for any r(E , j57')-quasi-complete locally convex space. In this paper we extend the result of Tweddle to set-valued measures. As in [6], we will use a theorem of E . C. James [4] which states that if X is a weakly closed subset of a r(E , Æ7')-quasi-complete locally convex space, then X is weakly compact iff each y in E ' attains its supremum on X . (Cf. also [5]
for a simplified proof of this theorem.)
Let E be a separated locally convex vector space over the reals with dual E'. Let $ be a set and X a a-algebra of subsets of 8. A set-valued measure M: E ->E is a mapping M which assigns to every element A in E a non-empty subset M(A) of E in such a way that M(A) = £ M (An), whenever A is the disjoint union of the sets An in E and £ M (An) — {æeE-, 1
n>l
there is a sequence (a?n)“=i such that oo= arLd æn€ M (A J for every n}.
«>i
We denote by <5*(*| A) the support function of a subset A of E : ô*(y I A) = sup «ж , у>: coeA} (yeE').
It is easy to prove that for every yeE' the mapping A->ô*(y \ M (A )) is a measure (whose values are in
] — oo, oo]).Th e o r e m.
Suppose E is r ( E , E')-quasi-complete and M : E -+E is a set-valued measure. I f the values of M are relatively o (E , E')-compact, then so is its range В — {M (A ): A eE}.
P ro of. For each fixed y *E ', à*[y | M (•)) is a finite measure on E
(see e.g. [1]). Let 8 = S + v 8 ~ be a Hahn decomposition of 8 with respect
to <5*(y I Ж(*)), so that 6*{y \ M(-)) is non-negative on 8 +, ô*{y \ M(-))
is non-positive on 8~ and 8 +r\S~ — 0.
402 E. Urbanski
Now, letting the bar denote the weak closure, we have sup{<a?,
2/> : xeR} = sup «а?, y>: xeR}
= sup{<
0?, y }: XeM (A), AeZJ]
= sup{<3*(y I M(A)): AeU}
= й*(у I Jf(fif+ )) = ô*(y I J f (#+)),
and since M {S+) is weakly compact, there exists an element xe M (S+) c: R such that ô* (y I M (S +)) — (as, yy. Thus, by the theorem of James men
tioned above, R is a {E , jE7')-compact.
R em ark . If M is a(E , jE/')-non-atomic (cf. [7]), then the weak closure of the range of M is convex (see [3]).
I am indebted to Dr. L. Drewnowski for his helpful remarks con
cerning the problem.
References
[1] Z. A rtste in , Set-valued measures, Trans. Amer. Math. Soc. 165 (1972), p. 103-125.
[2] E. G-. B a rtle , N. D u n fo rd and J. T. Sch w artz, Weak compactness and vector measures, Canad. J . Math. 7 (1965), p. 289-305.
[3] L. Drew now ski, Additive and countably additive correspondences, Comm. Math.
19 (1976), p. 25-52.
[4] E. C. Ja m e s, Weakly compactsets, Trans. Amer. Math. Soc. 113 (1964), p. 129-140.
[5] J . D. P ry ce, Weak compactness in locally convex spaces, Proc. Amer. Math.
Soc. 17 (1966), p. 148-155.
[6] I. T w ed dle, Weak compactness in locally convex spaces, Glasgow Math. J . 9 (1968), p. 123-127.
[7] — The range of a vector-valued measure, Glasgow Math. J . 13 (1972), p. 64-68.
IN S T IT U T E O F M A TH EM A T IC S A. M IC K IE W IC Z U N IV E R S IT Y POZNA N
