Real and Complex Analysis (Rudin)

48 REAL AND COMPLEX ANALYSIS
2.17 Theorem Suppose X is a locally compact, a-compact Hausdorff space. If
IDl and p. are as described in the statement of Theorem 2.14, then IDl and p. have
the following properties:
(a) If E E IDl and E > 0, there is a closed set F and an open set V such that
F c: E c: V and p.(V - F) < E.
(b) p. is a regular Borel measure on X.
(c) If E E IDl, there are sets A and B such that A is an F", B is a G 6 ,
A c: E c: B, and p.(B - A) = o.
As a corollary of (c) we see that every E E IDl is the union of an F" and a set
of measure O.
PROOF Let X = Kl U K2 U K3 U "', where each I\n is compact. If E E IDl
and E > 0, then p.(Kn n E) < 00, and there are open sets v" :::;) Kn n E such
that
E
p.(v" - (Kn n E» < 2 n + 1 (n = 1, 2, 3, ... ).
(1)
If V = U v" , then V -
E c: U (v" - (Kn n E», so that
Apply this to Ee in place of E: There is an open set W :::;) Ee such that
p.(W - £C) < E/2. If F = we, then F c: E, and E - F = W - £C. Now (a)
follows.
Every closed set F c: X is a-compact, because F = U (F n Kn). Hence
(a) implies that every set E E IDl is inner regular. This proves (b).
If we apply (a) with E = 1/j U = 1, 2, 3, ... ), we obtain closed sets F J and
open sets ~ such that F j c: E c: ~ and p.(~ - F J ) < 1/j. Put A = U Fj and
B = n ~. Then A c: E c: B, A is an F", B is a G." and p.(B - A) = 0 since
B - A c: ~ - F j for j = 1,2,3, .... This proves (c). IIII
2.18 Theorem Let X be a locally compact Hausdorffspace in which every open
set is a-compact. Let A be any positive Borel measure on X such that A(K) < 00
for every compact set K. Then A is regular.
Note that every euclidean space Rk satisfies the present hypothesis, since
every open set in Rk is a countabie union of closed balls.