We have
mf(R) = lim mf(R)
n
= lim(f(s',t') f(s ,t') f(s',t ) + f(s ,t )
n n n n
n
= f(s',t') lim f(s ,t') lim f(s',t ) + lim f(s ,t )
n n n n
s "ts t +tt s + s
n n n
t +tt
n
= f(s',t') f(s_,t') f(s',t ) + f(s_,t ).
With these in hand, to obtain the measure of other rectangles it is
simply a matter of adding or subtracting the appropriate intervals
that comprise the sides of the rectangle. We illustrate this
procedure (as well as check our work!) by using R and some intervals
to compute m (((s,t),(s',t')]). (Note that the rectangle
((s,t),(s',t')] (s,s'] x (t,t'].) We have, denoting this rectangle
by A (Figure 3-8),
(s,t ) (s',t')
4--------------
I
I A
I
(st)- :s',t)
Figure 3-8 The rectangle (s,s']x(t,t']