diIX(w) i
Taking M = [O,u], we obtain = , and
fMd IX(w)j = f[Ou].dlXI.(w); hence =
J[0,u].d[Xj (w). Putting this together with what we had
2
earlier for , we now have, for Z E S, u E R
U
O,u]. dJ (w) = J o dX .(w).
[0,u] w u [Ou] *
There is then a X(w)i-negligible set N (w,z)C R2 outside of which
we have = . In fact, the two integrals above form
measures on B(R2). By taking differences, we have, for u__diXi(w) = J(u,u]dXI (w), and these rectangles,
along with those [0,u] generate B(R ); hence =
pjX(w)i-a.e.
Now, the set N1(w) = UN (w,z) Is 1, X(w)-negligible, and for
u i N (w) we have = for all z e S; hence for all
z a Z since S is dense in Z. Since Z is norming, we have
H (w) = G (u) for u e N1(w).
U W
Let A = {(u,w): IHu(w)I < 11. A is then PX-measurable (in fact,
A = AOJ N with A E M, N PIXI-negligible. Then N is VX-negllgible
so A is VX-measurable). For each w, consider the section
A(w) = ulIH u(w)| < 1}; since for w t N UN1 U N2 we have
IGl = IIx(w) -a.e. and Hu(w) = Gw(u) PIx(w)l-a.e., we deduce that
A(w) = ju: IHu(w)I < 1 = (u: IGw(U)M < 1) (a.e.) is ,iX(w)-
negligible. Then uixI(A) = E(f 21A(w)(u)dIX u(w)) -' since
__