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A be a pIV -negligible set outside of which X is separably valued
and Xn X pointwise; then ,lvj(A) = E(f 2AdIVI z) = 0 >
f A1 (w)dlV (w) = 0 P-a.s. Denote the exceptional set by N. For
z
w i N, the section 1A(w) is dlVl (w)-negliglble, so
Xn(w) Xz(w) dlVl.(w)- almost everywhere, and XmXn(w) l IXz(w)I.
Now, since E(fIXz dIVIz) < -, there is a P-negligible set N1 F
such that for w t N fiXz(w) dVj|z(w) < -, i.e., IX(w)I is
d|lV(w)-integrable. Then for w t N U N we have
IX"(w)I < IX(w)I E L1(dIV(w)) and |Xn(w) IX(w) dlVl.(w)-a.e.
so by Lebesgue fR2 Xn(w)Idlviz(w) 4 fR21Xz(w)dlVlz(w) for
w e NUNI and in particular f 2X(w)dV (w) f 2X (w)dV (w)
R 2 z 2 z z
+
for w i N N1
Repeating the procedure, since the map w f R 21Xz(w) dlVlz(w)
is P-integrable, fIXn(w)d lV I(w) S fIXz(w)IdlVIz(w) P-a.s. and
f|Xn(w) IdlVI(w) f|Xz(w) d[VIz(w) P-a.s., we can apply Lebesgue
again and deduce that we also have convergence in L (P); in
particular, E(J 2X~~ dlvIz) E(f 2IX dlVIz). Moreover, for each n
R R
we have E(IfXndVz ) S E( IXnzdlV I) z E(fIX zdlV z) < ; hence
fX (w)dVz(w) c LF(P) for each n. (Note: We must show this map is
measurable; this will come out of a later computation.) We already
showed that fxn(w)dV (w) fx (w)dV (w) P-a.s. so by Lebesgue the
Z Z Z Z