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.