have If -fl < I => for n Z no we have Ifl | fnO + 1,
which is dg-integrable on I.) Similarly, f x fx
uniformly, so from some index n0 on, Ifn -fx| < =>
for n>n0, |fnx| < fnO x + 1 S fn0 1ixI + 1, integrable
since H is a vector space. By Lebesque, then,
ffnxdg fifxdg. Finally, since Igxl S Igl-Ix fn
are d(gx)-integrable, so fn d(gx) fJfd(gx) by Lebesgue as
before. For each n we have xJffndg = JIxfndg = IIfnd(gx),
so on passing to the limit we get xf fdg = f xfdg = f fd(gx),
which is (4.3.1).
iv) Let (f ) be a uniformly bounded increasing sequence of
n
positive functions from H, and denote f = lim f Show
n
n
f E H. Let M 2 f for all n. Then we have
n
fLfndlgl S fiMdlgl M-.gl(I) for all n. By Lebesgue, f is
dg-integrable on I, and Ifndg Jffdg, hence
xlfn dg xlfdg. Similarly, Ifnxl OfnIlxl
M M*-XI E L1(digj), so fx is dg-integrable on I by Lebesgue,
and f xdg f+ fxdg Also, fn is d(gx)-integrable since
Ifn] 5 M and fiMd(lgxl) 5 MxIJgIj(I), so by Lebesgue again