for the right hand side, JM<H,z>dp XI = E(J1M<H,z> dlxI ) (by Theorem

4.1.4, since I<H,z>I S IIHI-I zl z[) = E(1CAfEu]<H,z> dIX ), which

is the right hand side of (4.2.3), thus proving the equality.

 Now, since (4.2.3) holds for all A E F, there is a P-negligible

set (depending on u and z) N(u,z)C 0 outside of which

<Xu,z> = f[O,u]<H,z> vdIX. Since both sides of this equation are

right continuous, there is a negligible set N9x) = Q N(u,z)
 u rational

outside of which <Xu,z> = I[0,u]<H> dlX for all u E R2

 Let S be a countable dense subset of Z and set N1 = N(z);
 zCS
N is negligible. Also, since |XJm is integrable, there is a third

negligible set N2 outside of which IX1 (w) < m. Let, now, w I

NO U N1U N2 be fixed. The function X.(w) = X(w) is an EO-valued
 2
function defined on R having bounded va-iation IX(w)I = XI (w).

Then (Theorem 3.1.1) it determines a Stieltjes measure pX(w) on

B(R2) with finite variation IpX(w)I. By Theorem 3.2.2, we have

I X(w)I = PIX(w) Then by Theorem 1.5.7, there exists a function
 2
G : R+ Z' such that


 1") IG = -1 w-a.e.


 2") <Gwz> is u X(w)-integrable for every z E Z and


 <X(w)(M),z> = f<Gw,z>d (w) for M c B(R2). In fact,


taking f = 1M in 1.5.7(2), we obtain (as for (2')):
 N