of showing that, as u decreases to z', the variation on each of the
three rectangles R1, R2, and R3 vanishes. We shall give separate
proofs for R1 and R2, and the proof for R is identical to that
for R2. Note first of all, that for u' such that z' < u' < u, the
corresponding rectangles R', R2, R3 satisfy R1 C R R2 R2,
R C R3, so by Proposition 2.2.5(iv), VarRl(f) M Var (f), etc., and
1 1
so each of the limits lim Var (f), lim Var (f), lim Var (f)
u++z' 1 u++z' 2 u++z' 3
exists and is nonnegative.
a) Denote z' = (s',t') u = (p,r). We show that
lm Var[(s',t'),(p,r)](f) = 0.
r++t'
Assume not: then there exists a>O such that, for all u>z', we
have Var[zu](f) > a. Let u0 > z', E > 0. Denote u0 = (p0,r ).
Since Var[z',u > a, there exist partitions a : s' = ,0 < s, <
s0,2 < ... < sm = O' 'O: t' = t0,0 < tO,1 < .. < ton r0 such
that
i) z ( t )'Cs t fj > a
si a -
RBOoXO xo0 B CO BR Oo 0
R CI R C II R C III
(See Figure 2-5.) Since each of the three sums is less than the