In particular, we have
IAI (f)I > IA (f+)g E.
h h
Upon summing over h, we obtain
Var (f;P) = E A (f) > E |6 (f+) nc > a nc.
h h h
Now, denote J = ((p,r),(p',r')]: we can take p<1, and decreasing with
e, so for any c, we have Ih = I C [(p,r),(p'+1,r'+1)]. Denoting
this latter rectangle by K, we have K I for all I (in general, I
depends on e), so VarK(f) > Var (f) > a ne. e arbitrary =>
VarK(f) > a. Now, the collection Jh depends on a, but they all have
union equal to J, so we can repeat the above procedure for any a and
keep I C K. Thus, VarK(f) > a for any a => VarK(f) = + m, a
contradiction on our assumption of finite variation of f. Then a has
finite variation on 6.
4) o is inner regular on 6. We observe first that, from the
fact that the right-limit of f exists at each point of R2, it follows
2 2
that for each z E R e > 0, there exists z' c R with z0 such that for any points u,v>z, with
lu-zi If(z+) f(z')I 5 Then [f(u+) f(z+)I
S- E
f(u+) f(z')j + If(z') f(z+), I + E = EF