in two variables, we have two distinct definitions of "increasing,"
but our decomposition satisfies both.
2
Proposition 2.3.4. Let f: R -+ R have finite variation (again, in
the sense of Definition 2.3.1). Then we can write f = f f2, where
f and f2 are increasing in both senses of Definition 2.1.3, namely
a) for (s,t) S (s',t') we have f (s,t) S f (s',t') and
f2(s,t) f2(s',t')
and
b) for (s,t) < (s',t'), A[(st),(s t )] ) Z O and
A (s,t),(s',t')](f 2) 0.
2
Proof. For (s,t) c R+, set f (s,t) = Ifl(s,t), f2(s,t) =
f (s,t) f(s,t) = Ifj(s,t) f(s,t). In remark (4) following the
definition of Ifj (Definition 2.1.3), we showed that jfl is increasing
in both senses. We then have only to deal with f2.
a) Let (s,t) S (s',t'). We have
f2(s',t') f2(s,t) Ifl(s',t') f(s',t') (Ifl(s,t) f(s,t))
(Ifj(s',t') If (s,t)) (f(s',t') f(s,t)).
We shall show f(s',t') f(s,t) Ifj(s',t') Ifl(s,t). Denote by
R1 the rectangle [(O,t),(s',t')], by R2 the rectangle [(s,O),(s',t)]
(Figure 2-14). We have
f(s',t') f(s,t)
I |f(s',t') f(s,t)I