m ((s,s']xltl) = f(s',t) f(s',t_) f(s,t) + f(s,t)
mf((s,s')xjt}) f(s',t ) f(s ,t_) f(s,t) + f(s ,t_).
iii) We now compute the measure of certain rectangles in R
If we allow the possibility of each side being open or closed, this
gives 16 different rectangles, and we do not give explicit
computations for them all. We shall go into detail for only a few,
and indicate the procedure for the remainder.
First, we shall give the measure of an open rectangle
R = (s,s')x(t,t). We write R = R where R ((s ,t ),(s',t)]
Sn n n n n
n
with (sn,tn) (s,t), (n ,tn) tt (s',t') (see Figure 3-6).
--(s',t'O
!----------^--)
^n
Figure 3-6 Approximation of open rectangle
We have, then,
m (R) = lim m (R
SI n
mr(R) = lim mf(Rn)
= lim (f(s',t') f(s,t ) f(s ,t') + f(s ,t ))
n n n n n n n n
n