2
We return now to the situation with f defined on R right
continuous, with finite variation |fl, both extended by zero outside
the first quadrant. We have an important equality we shall make use
of in the next chapter, which is given in the following.
Theorem 3.2.2. Let m be the E-valued measure associated with f, and
let mlfi be the real-valued measure associated with Ifl. Then mf has
finite variation Imfl and we have the equality
ml Imflt
Proof. We showed in Theorem 3.1.1 that mf has finite variation. The
real thing to be proved here is the equality.
Let S be the semiring of rectangles of the form
R = ((s,t),(s',t')]. We shall show first that mlfI = Imfl on S.
We must consider various cases.
First of all, if (s',t') < (0,0), then mfl (R) = Imf (R) = 0.
We will assume, then, in what follows, that (s',t') lies in the first
quadrant. There are four cases, according to what quadrant (s,t) lies
in.
1) Assume (s,t) lies in the first quadrant. Let o: s =
s0 < s1 < ... < S = s' be a partition of [s,s'], T: t = t0 < t
< ... < t t' be a partition of [t,t']. Denote P = oXT the
corresponding partition of R:
P =R ijRij = [(st j),(s i+stj+)], 0 i S m-1, 0 5 j 5 n-l
Denote by R. the corresponding half-open rectangles
1,J