SR21A(w)(u)dlXlu(w) = 0 P-a.s. Then IHI = l- Ix-a.e.; hence by
(3') we have IPXI(M) = fMIHIdpXI = fMldI x = plx (M) for M E M,
i.e., |vX = P XI. The remainder of the theorem (D E L1 (P) iff
E(I 2u jdlXu! ) < ") will be proved in the next theorem in more
+
generality. I
Proposition 4.2.2. Let E,F be Banach spaces and V: R2xQ + L(E,F)
be a process with V integrable for every z E R2 and with raw
integrable variation IVI. If X is an E-valued measurable process,
then X E LE(~V) Iff X is ,Xi -almost separably valued and
E(Rf I2 IzdlVz) < -. In this case, E( RX zdV ) is defined, and
+ +
v (X) = E(fX dV ).
Proof. One way is easy: if X E LE(p ), i.e., X e LE IVi), then X is
plil-almost separably valued (being measurable) and E(f 2 XzdlVI )
< -. In fact, if X c LE(, Vl), then JXL( = pIVI( Ix) =
E L I viI
E(f R2 Xzdlvlz) by Theorem 4.1.4 (which holds for positive measurable
processes as well by monotone convergence), and this is finite by
assumption.
For the other implication, let X be an E-valued, measurable
process, puI -almost separably valued and satisfying the condition
E(f 2 Xz dVIz) < Let (Xn) be a sequence of u -measurable step
processes such that Xn X i V-a.e. and iXn X| for every n. Let