Var[(st t f) = Var[(s,t),(s.t.)]
+ Var[(st (s',t")]
(See Figure 2-2.)
1 S Si
S s' st
Figure 2-2 Additivity of the variation
iv) If R1 C R2, Var (f) S VarR (f)
1 2
v) If f is right continuous (order sense), we can compute
Var [zz'(f) using grids consisting of points with rational
E zz 7
coordinates (and hence we can take the supremum along a
sequence of partitions).
Proof.
i) Since Var [z,'(f;P) Z 0 for any partition P, we have
Var [zz'](f) = sup Var z,z'](f;P) 0.
[zz'] [zz']
ii) If Var [zz](f) = + -, then for every N>0, there is a
such that Varz,z By Lemma 2.2.2, there
partition PN such that Var .](f;P ) > N. By Lemma 2.2.2, there
-N [z,z'-N