for the right hand side, JMdp XI = E(J1M dlxI ) (by Theorem
4.1.4, since II S IIHI-I zl z[) = E(1CAfEu] 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
= f[O,u] 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 = I[0,u] 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") is u X(w)-integrable for every z E Z and
= fd (w) for M c B(R2). In fact,
taking f = 1M in 1.5.7(2), we obtain (as for (2')):
N