limit is P-integrable and we have E(fJXdV ) E(JX dV ). (in
z zz z
particular E(XzdVz) is defined). Next, we show that X E L (u )
for each n, uvi ( IX') = E(f IXnjdV z) S E(fjXz dIVjz) < -, so by
Fatou we have yVi (lim infIXn") S lim inf pli(( IXn) < m, in
particular lim inf |x"n is pV -integrable. But (XI = limlX"n
a.e. so IXI is puvI-integrable. Also, X = lim Xn is u -measurable,
n
so X E LE V). Moreover, u(Xn) + (X) by Lebesgue again, since
Xn X EL L (up i).
Finally, we show that, for each n, we have iV(X") E(fX dV ),
so we get the desired equality by passing to limits. Being a step
k
process, we can write Xn = E 1M Xi, Mi E M, X E E. Then we have
i=1 i
V (X") V(zI xi) = Ex iU(M ) = Exi(E(1 dVz)) (by Theorem 1.2.1)
1 i
= ZE(xiM dVz) = EE(Ml x dVz) (by Theorem 4.1.3) = E(f(E1M x )dVz)
= E(fXndV ) (and in particular the map fXn(w)dV (w) is P-
measurable). Letting n + m, we obtain u (X) = E(fx dV ), and the
theorem is completely proved.
Remarks.
1) By taking E = R in the statement, we have the following: If
X is any scalar-valued measurable process, then X E L (u ) iff