Il(s,t) xlzjgzjlg (s,t) < m
for all (s,t) E R2, i.e., has finite variation for all
x E E, z E F'.
Now to prove qualities (4.3.1) and (4.3.2): Let I B(R2)
f: R2 P R be dg-integrable. Assume Igl(I) < m. We shall use the
monotone class theorem 1.5.2 to prove the qualities for bounded f,
then extended to f-integrable.
Let H denote the set of bounded, real-valued, dg-lntegrable (on
I) functions f satisfying (4.3.1). Then:
i) H is a vector space (evidently).
ii) H contains the constants: let f = a constant. Then
xfifdg xfladg = xfac1dg = x(ag(I)) = (ax)g(1)
(g(I) is the measure of I for the measure dg),
flfxdg f(ax)lIdg = (ax)-g(I), and
f fd(gx) = falId(gx) = a-(gx)(I) = a(g(I)-x) = (ax)g(I).
Hence xf fdg = flfxdg = flfd(gx), which is (4.3.1).
iii) H is closed under uniform convergence: suppose f n f
uniformly and (4.3.1) holds for each f Then
n
xflfndg xflfdg, since by Lebesgue dominated convergence we
have that f is dg-integrable over I and f fndf + f1fdg,
hence x f ndg xi fdg. (In fact, from some Index n0 on we