Denote
R = (s,s'] x (t,t']
R2 = (s,s'] x (t',t"]
(Figure 3-1). (The proof is the same if R2 is of the form
(s',s"] x (t,t'].) We have
m(R1 R2) = f(s"+,t"+) f(s+,t"+) f(s'+,t+) + f(s+,t+)
= [f(s'+,t"+) f(s+,t"+) f(s'+,t"+) + f(s+,t'+)]
+ [f(s'+,t'+) f(s+,t'+) f(s'+,t+) + f(s+,t+)]
(we added and subtracted f(s'+,t'+) and f(s+,t'+))
= m(R2) + m(R ).
3) o has finite variation on 6. We prove this by contradiction;
suppose there exists a rectangle J E 6 such that Jol(J) = + w (oGl
denotes the variation of o). Then, for each a>O, there exists a
finite family (Jh), h = 1,2,...n of disjoint rectangles from 6,
JhC J for all h, such that
n n
E Jo(Jh ) > a, i.e., E ~6 (f+)| > a.
h=- h=1 n
Denote J = ((sh,th),(s th)] for all h. We may, of course, assume
n
that J = ) Jh (so that 'Jh e 6). Let > O. Since f has
n=1 h