therefore, consider the upper corner of A to be (s,t) for each n, so
that we can take An = ((pn,qn),(s,t)] without loss of generality. We
have, then, by o-additivity of m ,
m f((s,t)}) = lim f (An) = lim A (f)
n n n
= lim (f(s,t) f(pn,t) f(s,qn) + f( n,q ))
n
= lim f(s,t) lim f(pn,t) lim f(s,qn) + lim f(p nq
n n n n
since the individual limits exist by Theorem 2.3.3
f(s,t) f(s ,t ) f(s ,t ) + f(s_,t_).
If we note that by right continuity we have f(s,t) = f(s+,t+), we see
that the measure of a point is analogous to the measure of a "half-
open" rectangle, except we use the four limits to compute the measure
of a point.
ii) We next compute the measure mf of intervals of the forms:
s)} x (t,t'], {s) x [t,t'], {s) x [t,t'), {s} x (t,t'), and the
analogous "horizontal" intervals.
We begin with the closed interval I = {s} x [t,t']. We have
I = Rn, where R are rectangles of the form ((sn'tn),(s',t)]
n Rn n n n n n
n=1
with (s ,t ) ++ (s,t), (s',t') + (s,t'). As before, we can take
(s',t') = (s,t') for all n, so that R = ((s ,t ),(s,t')] with
n < s, tn < t, (s ,t ) tt (s,t) (see Figure 3-4).
Sn n