Real and Complex Analysis (Rudin)

244 REAL AND COMPLEX ANALYSIS
lim sup UO(ZJ) :::; E/c". (6)
j-+ 00
Since U = Uo + U 1 and E was arbitrary, (4) and (6) give
lim u(z J) = O. (7)
j~oo
11.23 Theorem If f E Ll(T), then PU] has nontangential limit f(ei~ at every
Lebesgue point ei8 off.
PROOF Suppose ei8 is a Lebesgue point off. By subtracting a constant fromf
we may assume, without loss of generality, thatf(ei8) = O. Then
IIII
(1)
as the open arcs I c: T shrink to their center ei8. Define a Borel measure J1. on
Tby
(2)
Then (1) says that (DJ1.)(ei8) = 0; hence P[dJ1.] has nontangentiallimit 0 at ei8,
by Theorem 11.22. The sam~ is true of P[f], because
I P[f] I:::; P[ If I] = P[dJ1.].
(3)
IIII
The last two theorems can be combined as follows.
11.24 Theorem If dJ1. = f d(1 + dJ1.. is the Lebesgue decomposition of a complex
Borel measure J1. on T, where f E Ll(T), J1. • .1 (1, then P[dJ1.] has nontangential
limit f(l!i~ at almost all points of T.
PROOF Apply Theorem 11.22 to the positive and negative variations of the
real and imaginary parts of J1.., and apply Theorem 11.23 wf. IIII
Here is another consequence of Theorem 11.20.
11.25 Theorem For 0 < ex < 1 and 1:::; p :::; 00, there are constants
A(ex, p) < 00 with the following properties: