stochastic measures X with finite variation Ixi given by the
equality
(4.4.1) u (A) = E(f A dX ) for A bounded, measurable.
2z z
Remark. We shall later prove the equality iXI = I|XJ for X wiin
values in a Banach space, from which the equality follows for real-
valued X as a special case.
Proof. We remark first that the correspondence is one-to-one in the
sense that we identify processes that differ only on an evanescent
set.
1) Let X: R2 x 4 R be a raw process with raw integrable
variation IXI. For any bounded measurable process A, Al [ M, the
map w 2A z(w)dXz(w) is in L1(P). In fact, we have
f 2Az(w)dXz w) R S I 2Az(w) di xz(w) S Mlxi (w), and by assumption
IXi E L1(P). Then, for any M E M = B(R2)xF, E(J 21MdX ) exists.
Set, then, ~X(M) = E(f 21 dXz). Then, for step functions
n n
B a ., Mi E M, aI e R, we have X(B) = JBdu = E a(Mi) =
i=1 1 i=1
n n n
E a(E( 2( )zdXz)) = E E(f 2 (a11M. ) dX = E( E ( 2(ai1 ) dX ))
i=1 R i i=1 BR i i=1 R 1
n
= E(f 2( E a M )zdXz) = E( 2B dXz). Now, suppose A is bounded,
R i=1 i R