Now for (ii). For x E E, z0 c Z we have =
a.s., so by (2') ** is integrable; moreover, is right
continuous since B is, and has Integrable variation (we showed in the
proof of 4.1.3 that ](w)l S IB|(w)|x|Dz o). Also, for each
2
x cR **** is F-measurable since B is; hence **** is a
+ z 0 z z 0
real-valued raw process with Integrable variation, which is (ii). We
now turn to assertions (1)-(4).
Proof of (1): For M = [O,z]xA, A E F, we have, for x,zo:
= = E(1A**** (by (2')) = E(1A)
S00 A Z 0 A Z
E(Af 21[0,z]d) = E(f Md). Then =
+
E(fl Md****)I E(fiMd dB 1x10Z0 ) = E(f 2 1MdIBI )MIzIOll
so .m(M)l 5 E(flMdlBl ); hence Iml(M) a E(l Md BI ). On the other
hand, from IBI S V we have E(J Md BI ) S E(fldVw) = Iml(M); hence
ImJ(M) = E(flMdIBl ) for M = [O,z]xA. By additivity, then, as usual,
we have Iml(M) = E(f Mdl BI) =- IB (M) for M = RxA, A E F, R a
rectangle of the semiring generating B(R2) (there are four kinds; cf.
Theorems 3.2.2 and 4.2.1). As Iml, flBI are both a-additive, and
sets of the form M = RxA form a semiring generating M, we have
Now, let X be E-valued, measurable. If X E L (m), then
X| c L (Iml) LI (1 I) => X LE(p iB). Conversely, if
X L i( ,), then IX E L (p BI ) => XI ec L (I l); hence
X E LE(m) since X is measurable.
E
**