Real and Complex Analysis (Rudin)

COMPLEX MEASURES 129
The first part of the proof shows now that there exists G E IJ(jl) such that
'I'(F) = 1 FG dil
(F E I!(il)). (10)
Put 9 = wl/qG. (If p = 1, 9 = G.) Then
11 9 Iq dJl = 11 G Iq dil = 11'I'll q = 1III q (11)
if p > 1, whereas Ilglloo = IIGlloo = 11'1'11 = 1111
since G dil = wl/Pg dJl, we finally get
if p = 1. Thus (2) holds, and
for every IE I!(Jl).
(!) = 'I'(w -lip!) = 1 w -I/PfG dil = Ilg dJl
(12)
IIII
6.17 Remark We have already encountered the special case p = q = 2 of
Theorem 6.16. In fact, the proof of the general case was based on this special
case, for we used the knowledge of the bounded linear functionals on L2(Jl) in
the proof of the Radon-Nikodym theorem, and the latter was the key to the
proof of Theorem 6.16. The special case p = 2, in turn, depended on the
completeness of E(Jl), on the fact that E(Jl) is therefore a Hilbert space, and
on the fact that the bounded linear functionals on a Hilbert space are given
by inner products.
We now turn to the complex version of Theorem 2.14.
The Riesz Representation Theorem
[i.lS Let X be a locally compact Hausdorff space. Theorem 2.14 characterizes the
Dositive linear functionals on Cc(X). We are now in a position to characterize the
;ounded linear functionals on Cc(X). Since Cc(X) is a dense subspace of Co(X),
relative to the supremum norm, every such
has a unique extension to a
bounded linear functional on Co(X). Hence we may as well assume to begin with
that we are dealing with the Banach~space Co(X).
If Jl is a complex Borel measure, Theorem 6.12 asserts that there is a complex
Borel function h with I h I = 1 such that dJl = h d I JlI. It is therefore reasonable to
iefine integration with respect to a complex measure Jl by the formula
f I dJl = f jh d I JlI· (1)
The relation J XE dJl = Jl(E) is a special case of (1). Thus
1 1
XE d(Jl + A.) = (Jl + ,1,)(E) = Jl(E) + A.(E) =
XE dJl + Ix XE d,1, (2)