hence m is the measure associated with the stochastic function
B": R2xn L(E,FO). Since F0 is separable, there is a countable
subset SC Z norming for FO. By (2), we have Iml(M) = E(f MdIB"I ).
Since B = B" outside an evanescent set, IBI = IB"i outside an
evanescent set, so I 21MdIBI 21 MdB"I a.s. for any M E M;
B +
hence Iml(M) = E(f MdJB"J ) = E(l MdIBJ ) for M E M, i.e.,
Iml = UIBi, and this completes the proof. I
Remark. If we start with a stochastic measure and associate a
function, we always have Iml = pUB but if we start with a stochastic
function, we do not get equality--not even if the measure has values
in L(E,F). Equality (1.1) seems to be as close as we can come in
general; in order to get everywhere from there, it seems we need for E
and Z not to be "too large."