converging to f a.s. and in L (d gl) on I, with (4.3.1) satisfied for
each n. In the first integral, we have 1f ndg '+ ffdg, hence
xf fndg 4 xf fdg. Also, Iflxfndg fxfdgR = If|x(fn-f)dg|
Sx IfIlfn-fldlgI 0; hence xj fndg xj fdg. Finally, f is d(gx)-
integrable [6, Theorem 4, p. 172], and we have IJifnd(gx) ffd(gx)|
= l1(fn-f)d(gx)| 5 flfn-fld(|Iglxl ) = Ix IIlfn-fld(jg|) 0 as
n 4 -. Then fifnd(gx) -* ffd(gx). Since (4.3.1) holds for each n, we
have it for f as well by passing to the limit.
Finally, let I e B(R2), f dg-integrable on I. There exists a
sequence (I ) of sets from B(R2) with Igl(I ) < m, and I t I.
1n n
Then f-1 f-1 a.s. and in L (djgl) and L1 (d( g lx[)) by Lebesgue
n
(since |fl1 S I f.1 I E L (dlgl) and L (djgf Ix )); (4.3.1) is
n
satisfied for each In, so we pass to limits exactly as above. This
completes the proof of (4.3.1). The proof of (4.3.2) is completely
analogous (since |< Sx jxzjIgl). I
We conclude this section with the theorem establishing the
correspondence between stochastic measures (P-measures) with finite
variation and processes of integrable variation for the case of real-
valued processes and measures. Although this is a special case of the
more general result we will establish later, it cannot be deduced from
that, since our proof for the vector-valued case will make use of the
real-valued result.
Theorem 4.1.4. There is a one-to-one correspondence X pX between
raw processes X: R x R with raw integrable variation jXI and