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 <gx,z>: R2 + R have finite variations gxl

and |<gx,z>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<gx,z> integrable on I, and we have


(4.3.1) xf fdg = f fxdg = fifd(gx)


and also


(4.3.2) <f fxdg,z> = <fifd(gx),z> = fifd<gx,z>.


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