4) A = [0,s'] x [O,t']: we get E(f AxFdXz) = E(1X(s.t.))
(Note: See Theorem 3.2.2 and preceding example for computations of
the measure dX(w) on these rectangles.)
By the Monotone Class Theorem, H contains o(U) = M, so E( 1MdXz
is defined for all M E M. Set p (M) = E(f 21MdXz). Then X: M + E
is a o-additive stochastic measure: X is evidently additive. If
M n 4, then for each w, f1 (w)dX (w) + 0 by o-additivity of the
n M z
n
integral. Then E(flM dXz) 0 by Lebesgue; hence pX Is o-additive.
n
Also, if M is evanescent, then M(w) is empty P-a.s. => flM(w)dXz(w) = 0
a.s. => pu(M) = E(f1MdXz) = 0.
Now, p also satisfies BiX(M) p IX(M), since u(M)I =
IE(l MdXz)I S E( JlMdXz ) E(fMdlX z ) = Ixl(M) ('vXI is the measure
associated with IXI by Theorem 4.1.4); hence iX has finite variation
|XI 5 p Xi (since the variation is the smallest positive measure
bounding the norm). We shall prove this is an equality.
Each Xz, being measurable, is almost-separably valued, so we can
find a common negligible set N0 outside of which Xz is separately
valued for z rational. By right continuity, X = lim X
z u
u+z
u rational
so for w e N X takes on values in a separable subspace
o
EOC E. Let Z C E' be a separable subspace norming for EO. Since
|uXI 5 pjiX which is finite, we have 1X << ulXI. By the extended
Radon-Nikodym Theorem (Theorem 1.5.8), there exists a stochastic
function H: R2xa Z' (=L(R,Z')) having the following properties: