Real and Complex Analysis (Rudin)

130 REAL AND COMPLEX ANALYSIS
whenever p. and A. are complex measures on IDl and E E IDl. This leads to the
addition formula
which is valid (for instance) for every bounded measurable!
We shall call a complex Borel measure p. on X regular if I p.1 is regular in the
sense of Definition 2.15. If p. is a complex Borel measure on X, it is clear that the
mapping
is a bounded linear functional on Co(X), whose norm is no larger than I p.1 (X).
That all bounded linear functionals on Co(X) are obtained in this way is the
content of the Riesz theorem:
6.19 Theorem If X is a locally compact Hausdorff space, then every bounded
linear functional Cl> on Co(X) is represented by a unique regular complex Borel
measure p., in the sense that
for every f E Co(X). Moreover, the norm ofCl> is the total variation of p.:
(3)
(4)
(1)
11Cl>1I = I p.1 (X). (2)
PROOF We first settle the uniqueness question. Suppose p. is a regular
complex Borel measure on X and J f dp. = 0 for all f E Co(X). By Theorem
6.12 there is a Borel function h, with I h I = 1, such that dp. = h d I p.1. For any
sequence Un} in Co(X) we then have
I p.1 (X) = 1 (h - fn)h d I p.1 ~ 11 h - Inl d I p.1 , (3)
and since Cc(X) is dense in Ll( I p.1) (Theorem 3.14), Un} can be so chosen that
the last expression in (3) tends to 0 as n --+ 00. Thus I p.1 (X) = 0, and p. = O. It
is easy to see that the difference of two regular complex Borel measures on X
is regular. This shows that at most one p. corresponds to each Cl>.
Now consider a given bounded linear functional Cl> on Co(X). Assume
1ICl>1I = 1, without loss of generality. We shall construct a positive linear functional
A on Cc(X), such that
I Cl>(f) I ~ A( I f I) ~ II f II (4)
where IIfll denotes the supremum norm.