Xn(w)dB (w) + JXv(w)dB (w). Then, by continuity, for w V N, z t Z,
we have < 2X (w)dB (w),z>; moreover,
II zlIfXn (w)dB (w)l IZIf Xn(w) dlBv(w) 5
AZlfJX (w) IdlB (w). Now, the function w IX (w)ldlBI (w) is P-
+
integrable, so by Lebesgue < 2X (w)dB (w),z> is P-integrable and
V+ V
E((< 2X"(w)dB (w),z> E(< X (w)dB (w),z>) for all z E Z. For each
H v V v v
n, E() as we saw above. Finally, X is
+
M-measurable by assumption, and jX E L (pIBJ) C L1 (ml); hence
X e LE(m), and Im(Xn) m(X)I S Iml( Xn XI) 0 by Lebesgue (since
jXnj r [Xi, Xn X ml-a.e.), i.e., m(Xn) m(X). Then *
for all z e Z. Passing to limits, we obtain =
E() for all z E Z, which completes the proof of (1).
Ad (2): Suppose, now, that Bx is separably valued for every
x E E. Let X E L( ), let Xn be step processes converting to
X JBI-a.e. with Xn 5S XB| for all n. We shall show first of all
that the map w 4 2Xf(w)dB (w) is integrable for all n; write
n
n
X = xi, Mi M disjoint, x. e E. For each i, Bxi is separably
i=1 1
valued. Also, by (ii), is measurable for z E Z. Since Z is
norming, Bxi is weakly measurable, so Bxi, being separably valued, is