Real and Complex Analysis (Rudin)

ABSTRACT INTEGRATION 31
PROOF Let a be a closed circular disc (with center at IX and radius r > 0, say)
in the complement of S. Since SC is the union of countably many such discs, it
is enough to prove that p.(E) = 0, where E = f - 1(a).
If we had p.(E) > 0, then
I A~f) - IXI = p.(~) I 1(f - IX) dp.1 ~ p.(~ 1 If - IXI dp. ~ r,
which is impossible, since A~f) E S. Hence p.(E) = O.
IIII
1.41 Theorem Let {Ek} be a sequence of measurable sets in X, such that
00
L p.(EJ < 00. (1)
1:=1
Then almost all x E X lie in at most finitely many of the sets E k •
PROOF If A is the set of all x which lie in infinitely many E k , we have to
prove that p.(A) = O. Put
00
g(x) = L XEt(X) (x E X). (2)
1:=1
For each x, each term in this series is either 0 or 1. Hence x E A if and only if
g(x) = 00. By Theorem 1.27, the integral of g over X is equal to the sum in
(1). Thus g E I!(p.), and so g(x) < 00 a.e. IIII
Exercises
1 Does there exist an infinite a-algebra which has only countably many members?
1 Prove an analogue of Theorem 1.8 for n functions.
3 Prove that iffis a real function on a measurable space X such that {x:f(x) ~ r} is measurable for
every rational r, thenfis measurable.
4 Let {an} and {bnl be sequences in [-