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 ** 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****) = E(I1Md****) = ;
M v M v 0
**