n n n
S1 <(M =)x ,z> = E E( 21 M.d**) = E E(< 21 dB x z>) (by
i=1 i=1 1 1-1 1
n
Prop. 4.1.3) = E( E < 1M dB xi,z>) = E(<(f1M dBvxi,z>)
i=1 1 1
n
E() (again by 4.1.3) = E(<( Z 1 x )dB ,z>) =
1 i=1 I
E(). Now, let X c Li (l1i) and let Xn be a sequence of
measurable step functions such that Xn + X pi -a.e. and |xnj S XII
everywhere. Let A be a p -negligible set outside of which X is
separably valued, fJX vldBI < m (more precisely, for (w,u) i A,
[O,u]Xv(w)ldlBlv(w) < ; since E(f 2Xv dlBIv) < ,
f 2lXv(w) dIBI(w) < P-a.s.; hence f[0,uIX (w) IdBI v(w) <
+
PiBI-a.e. since it is a P-measure), and X" X. There then exists a
P-negligible set N E F such that for w t N, the section A(w) is
dlB|.(w)-negligible (in fact, E(I1A (w) diB (w)) = pVBI(A) = 0 =>
fR2 A(w)dBI.(w) = 0 P-a.s.), so for w e N we have d|B .(w)-a.e.:
i) X (w) is separably valued
i ) |Xn(w), s IX.(w)l
111) xn(w) 4 X (w).
Since 2X v(w) ldIBI(w) < ( for w i N, X.(w) is dlBl.(w)
integrable, and Xn(w) + X (w) in L (djBI.(w)), so by Lebesgue
**