and
Iml(jXjj) E(JXIu jd BIu).
2) If, in addition, Bx is separably valued for every x E E and
if X is u, -integrable, thnthe integral E( X udB ) is defined, and
2 u
m(X) E(f XudBu).
3) The measure m has values in L(E,F) in each of the following
cases:
a) F Z'.
2
b) For every x c E and v c R, the convex equilibrated
(balanced) cover of the set B3 (w)x: w C n1 is
relatively o(F,Z)-compact in F.
c) For every x E E and v R2, the function 3 x is F-
+' v
measurable and almost separably valued; in particular,
this is the case if F is separable.
Proof. Let p1BI be the measure generated by jBI via Theorem 4.1.4.
For every x E E and z E Z the variation of the process
satisfies [I 5 Bl x lzl (cf. proof of Prop. 4.1.3).
Let mz be the stochastic measure generated by :
m z(M) = E( R21Md) for M e M. The mapping (x,z) m (M)
is linear in each argument: in fact, mx1+x2,(M) =
E(f 21Md**) = E( 1Md) = E(f Md(+))
H
**