Denote u2 = (p2,r2) (s1,1't,1). By assumption,
Var[r ,u2 (f) > a, etc. Continuing in this manner, we construct a
sequence u0, u u2, ... with ui > ui+1 > z' for all i such that, for
all i, the total variation of f on the rectangles making up
[(s',t'),(pO,r0)] \ [(s',t'),(Pi,ri)] is greater than
i
(a -) + (a ) + ... + (a 21 ia ia E,
2 4 2 j=1 2
hence we have Var [z',u(f) > ia E for all i, i.e.,
Var z,u(f) = + -, a contradiction of the hypotheses that
Var (f) < on every bounded rectangle. Hence, for any aO0, we have
0 lim Var, (f) < a => lim Var (f) = 0.
pts' uz' 1)(p)
r+4t'
This takes care of R1.
b) We show now that lim Var (f) = 0, or, more precisely, that
R
u+z' 2
lim Var[(s t(f) = 0. (See Figure 2-6). We proceed as
P +S [(s',t),(p,t')]
before, by contradiction. Assume there exists a>0 such that
(s,t') z/ (p,t')
R2
z(s,t) (s',t) (p,t)
Figure 2-6 Computing variation of R2