distinct definitions of "increasing." However, we have shown (Prop.
2.3.4) that a process of finite variation, as we have defined it here,
can be written as a difference of two processes (apply Prop. 2.3.4 to
each path) that are increasing in both senses, thus removing the
ambiguity.
We give now one more result concerning functions, which will be
used extensively in later theorems.
Proposition 4.1.3. If g: R2 L(E,F) is a function with finite
variation Igl (Defn. 2.3.1), then for every x E E and z E F', the
2 2
functions gx: R F and : R2 + R have finite variations gxl
and |l. (For the real-valued functions we shall use double bars
for the absolute value to avoid confusion.) Moreover, if f: R2 R
2
is dg-integrable (i.e., dlgl-integrable) on a set IC R2, then f is
d(gx)- and d integrable on I, and we have
(4.3.1) xf fdg = f fxdg = fifd(gx)
and also
(4.3.2) = = fifd.
Proof. For the first assertion, let z = (s,t) E R We have, from
Definition 2.3.1,
Igl(s,t) = Ig(0,0) + Var[ ,]g(.,O) Var 0,t]g(O,.)
ls1(9~, %(001 BPI O's I ~ t
+ Var[(0 st)] < .
E(o,o),(s,t)J