indistinguishable; hence IB'| = IBI up to evanescence, so


l1B = P18'I = Iml.
 2
 1) Assume p[B ] = B for all z R We have, from above,

 p[B'] = p[B ] for z E R as well. Then from (1.1) we conclude that

 B = B' a.s. for each z (property 4 following Defn. 1.5.5). Since
 z z
 both are right continuous, they are indistinguishable.

 2) Assume E is separable, let E C E be a countable dense set,

 S CZ a countable subset norming for F. We have <B'x,z
 z 0
 <B x,z0> a.s. for all x E EO, z0 E S. There is then a common

 2
negligible set N such that the equality is valid for all z E R

rational, x e EO, z0 E S. By right continuity, then, this holds
 2
outside N for all z E R Since S is norming, we have B'x = B x
 + z z
 2
outside N for all z R x E EO. Since E is dense in E, we have
 P
B'x = B x for all z E R, x c E (still outside N); hence B' = B
z z + z z
outside the evanescent set R xN, i.e., B and B' are indistinguishable.

 3) Assume now that E is separable and B x is integrable for
 2
every x E, z E R Then B x is almost separably valued; by right

continuity Bx is separably valued outside an evanescent set for

x c E. Since E is separable there is (as before) a common evanescent

set A outside of which Bx is separably valued for all x e E. We

modify B on A by setting it equal to zero on A and get a process B"

indistinguishable from B, with B" taking values in a separable space
 x
FO C F for all x c E. Then


E(f1Md<B"x,z>) = E(I1Md<B x, >) = <m(M)x,z>;
 M v M v 0