Ad (3):
(a) is trivial.
(b): Let x e E, veR Since the set C = co(Bv (w)x: wean
(balanced closed convex hull) is o(F,Z)-compact, the natural embedding
of C in Z*, the algebraic dual of Z, is o(Z*,Z)-compact (see Dunford and
Schwartz [8]). There is then a family (Zi.)EI of elements of Z such that
C = (tyEZ*: cI 1i (any closed convex set Is an intersection
IEI
of half-planes; we can use balls since C is equilibrated). Then we
have I**I 5 1 for all i E I, w E Q. Let M = [0,u] x A,
A E F; we have II = IE(Jflou]xAd****) =
IE(1AI S E(1AI ****) 5 1; hence m([O,u]xA)x c F,
i.e., m([O,u]xA) E L(E,F). By taking differences, we have
m((u,u']xA) E L(E,F), and also finite disjoint unions of such sets.
We shall use the monotone class theorem to prove that m(M) c L(E,F)
for all M E M.
Let M = IM E M: m(M) E L(E,F)]. We show first that is a
monotone class: Let M E M M + M. Then m(M )x E FC Z'. We have
n n n
Mm(Mn)x m(M)xI 5 Im(Mn) m(M) Ixl + 0 by a-additivity of m. Hence
m(M )x + m(M)x in the metric topology of Z'. Since F is closed in
Z' for the metric topology, m(M)x E F as well, i.e., m(M) E L(E,F).
The proof for M + M is exactly the same.
Now, let C be the algebra generated by sets of the form
(u,u']xA, A E F. C consists of finite unions and complements of such
sets. We have shown that if M is a finite union of such sets, then
**