v)
f(s,t) =
rl
q s s' q'
Figure 2-3 Decomposition of R2
Suppose, now, f is right continuous, i.e.,
lim f(s',t'). We show first that, for any grid Q = oxT
s'"s
t'+t
on R 00, there exists a grid Q' = o'xT' with rational coordinates*
such that
I 1 IRn fi E Z jfll < .
R EQ 1 R EQ' j <
Let, then, R [(s,t),(x,y)], a: s = s0 < s < ... < s = x be a
partition of [s,x], T: t = tO < t, < ... < tn = y be a partition of
[t,y]. We can choose points s5 > so, s; > a ,..., s > s and
t0 > to, t > t1... t' > tn so that, for each 0 5 k S m, O 5 1 < n,
we have If((sk t) ,t)f(s < k -- We take then
o': s9 < < ... < and T': t < t' < ... < t'. Then for each
rectangle Rk [(s t ) (sk1 ,t +)] E Q, there corresponds a
* Note: Q' is not actually a partition of R, but here f is defined
outside of R so we can use these sums to compute the variation.
The main point of this is to show the variation is the limit of a
sequence, which we shall need later on.