2) The theorem holds in particular for the situation we use:
2
that where f is defined on R with If| bounded, and f extended by
zero outside the first quadrant.
2
Suppose, now, that f is defined on R right continuous, with |f]
finite, and extend f by zero outside the first quadrant. As an
exercise, we shall compute explicitly the measure of some sets in R2
using the limits developed in Theorem 2.3.3. More precisely, we will
compute the measure of points, intervals, and some rectangles in terms
of the "quandrantal" limits of f.
2
i) Let (s,t) E R Denoting by mf the measure associated
with f, we compute mf ((s,t))). We can write {(s,t)} ( An'
n=1
where A denotes the rectangle ((p ,q ),(p',q')] with
(pn,qn) < (s,t) < (pn,qn), pn ++s q n+', pn's, q'+t (see Figure
3-3). If we decompose A into four parts, labeled I-IV in Figure 3-3,
we see that as (pn,q') + (s,t), parts I, II, and IV vanish. We may,
An
(p',q )
n n
I i
III (s,t)
nIV
I IV
(P ,q0)
Figure 3-3 Approximation of {(s,t)J