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']