Real and Complex Analysis (Rudin)

46 REAL AND COMPLEX ANALYSIS
PROOF The countable additivity of JI. on 9Jl follows immediately from Steps
IV and VIII.
IIII
STEP X For every f E Cc(X), Af = fx f dJl..
This proves (a), and completes the theorem.
PROOF Clearly, it is enough to prove this for real f Also, it is enough- to
prove the inequality
(16)
for every realf E Cc(X). For once (16) is establi!1hed, the linearity of A shows
that
which, together with (16), shows that equality holds in (16).
Let K be the support of a real f E Cc(X), let [a, b] be an interval which
contains the range off (note the Corollary to Theorem 2.10), choose E > 0,
and choose Yi' for i = 0, 1, ... , n, so that Yi - Yi-l < E and
Put
Yo < a < Yl < ... < Y .. = b. (17)
Ei = {x: Yi-l Ei such that
E
JI.(V;) < JI.(EJ + - n
(i = 1, ... , n) (19)
and such thatf(x) < Yi + E for all x E V;. By Theorem 2.13, there are functions
hi < V; such that L hi = 1 on K. Hence f = L hJ, and Step II shows
that