E(fJIXzdVl z) < -. (X is automatically separably valued.) Then
E(ix dV ) is defined, and uv(X) = E(IX dV ). Finally, equality
(4.1.2) is proved the same way as (4.1.1), by taking step processes
and passing to limits.
2) The correspondence is not one-to-one: as we shall see in the
next section, a stochastic measure with values in L(E,F) is generated
by a stochastic function(not necessarily measurable) with values in a
subspace of L(E,F").
We can also generate stochastic measures from stochastic
functions (not necessarily measurable) with raw integrable variation,
as the next theorem shows.
Theorem 4.2.3. Let E,F be two Banach spaces and ZC F' a subspace
norming for F. Let B:R xQ L(E,F) be a right-continuous stochastic
function satisfying the following conditions:
i) B has raw integrable variation [BI.
ii) For every x E E and z E Z, is a real-valued process
(measurable!) with raw integrable variation Il.
Then there exists a stochastic measure m: M 4 L(E,Z') with finite
variation Im| satisfying the following conditions:
1) If X is an E-valued process and if X is vIB -integrable, then
X Lm), the integral E(< X (w)dBu(w),z>) is defined for every
z E
z E Z'
= E(),