110
E(1 X ). On the other hand, E( ) E(1Baz ) E (llim a ))
B B Zz+ 3 U
urZ
= lim (E(1 a )) (by monotone convergence) < lim (E(1 a)) =
uz B u Uz B
uiz uiz
lim A (B) = A (B) by right continuity. Putting the two together,
u+z
A (B) = E(CXz) for B e F, i.e., Xz is a density for Az
Then for processes of the form A (w) Y(w) [I ,u(z),
u(A) = E( R2AzdXz), since E(J 2AzdXz) = E(R 2Y(w) Ir,u](z)dX
4 + +
= E(Y(w)f RI dX) = E(Y(w) X )= (Y) = p(Y I ) = u(A).
2[,u] z u u [G,u]
1 1 1
Moreover, az+ lim a so az+ is also (same proof)
u+z
u rational
Incrementally increasing for w e N; hence X = a1 is an integrable
z z+
Increasing process: more precisely, l(s,t)(w) X(0,0)()
(s (w) + (0,t) + (st)(w) 4X (s,t)(w) for w e N, so outside
an evanescent set IXI 4 i(R2xn) (we shall use this later). To prove
the equality (4.4.1) for A bounded, we use a monotone classes
argument.
Let H be the set of all bounded, M-measurable processes A (w) for
Z
which (4.4.1) holds. H is clearly a vector space. Also, H contains
the constants: If A = c constant, then A = lim c I (z),
n+>. [(0,0) ,(n,n)]
n-m
and we have shown (4.4.1) for these processes already. We get the
result in the limit by monotone convergence. More precisely, denoting
A (w) = c I n)(z), we have, for w outside a negligible
z s (0,O),(n,n)=
set, supf An(w)dX (w) = supX (w) < -, so by monotone convergence
2 z (n,n)
I