x c E. This amounts to showing that <m(MJ N)x,z> = <m(M)x + m(N)x,z>

for all x E E, z e Z. Now, <m(MUN)x,z> = E( 21 M d<B x,z>)
 -2 MUN v

 SE( R2(1 M+N)d<B x,z>) = E(R 2lMd<BvX,z> + R21Nd<B ( Z>) =


 E( R21 Md< x,z>) + E(f 21Nd<B x,z>) = <m(M)x,z> + <m(N)x,z> =
 + +

 <m(M)x + m(N)x,z>, 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 <m(X),z> = <m( M x ),z> = < E m(M )x ,z>
 i=1 1 i=1