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]).**

**T****h 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