x c E. This amounts to showing that =
for all x E E, z e Z. Now, = E( 21 M d**)
-2 MUN v
SE( R2(1 M+N)d****) = E(R 2lMd + R21Nd****) =
E( R21 Md< x,z>) + E(f 21Nd****) = + =
+ +
, which shows that m is additive. If, now,
An +0, jm(An) ) 0 since Im(An)I S hIBI(An) and the latter is
a-additive. Then m is a-additive as well.
As for the variation, i BI is a bounded, positive measure
satisfying Iml uIBI ; hence Iml S V\B\ since the variation is the
smallest positive measure dominating the norm. In particular, m has
finite variation.
Now we prove assertions (1)-(3).
Ad (1): Let X be an E-valued process. From the inequality
Iml 5 pB it follows that if X e LE(1IBI), then X E Ll( m )
(since any sequence of step functions Cauchy in LE wB| is then also
Cauchy in L (Iml); hence X E L (m) since X is M-measurable, and
iml ( X) S UJlB(jIX) = E( R 2Xl vdBl ), which is the second part of
(1).
The first part of (1) is satisfied for any M-measurable step
n
process X = 1M xi, Mi E M disjoint, xi E E. In fact, for
i-i 1
n n
z E Z, we have = = < E m(M )x ,z>
i=1 1 i=1
**