3) We can choose B with values in L(E,F) in each of the
following cases:
a) F is the dual of a Banach space H and we choose Z = H;
hence F = Z'.
b) For every x c E, the convex equilibrated cover of the set
{f(xdm: t simple process, Jl|Idlml 1} is relatively o(F,Z)-
compact in F.
c) E is separable and F has the Radon-Nikodym property (we
say F E RNP); in this case B can be chosen such that Bx is measurable
and separably valued for every x e E, hence
m(X) = E(fX dB ) for X E L (m).
d) The range of m is contained in a subspace G L(E,F)
having the RNP: in this case B can be chosen measurable, with
separable range contained in G; hence
m(*) = E( 4udB ) for Ec L (m).
4) If p is a lifting of P, we can choose B uniquely up to an
evanescent set, such that p[B ] = B for every v R2 (see Definition
V V +
1.5.5(b)).
Proof. Let V be the integrable increasing raw process associated with
Iml via Theorem 4.1.4:
ml(M) = E(f 21MdVu) for M c M.
+H