Real and Complex Analysis (Rudin)

146 REAL AND COMPLEX ANALYSIS
for any n and any disjoint collection of segments (lXI' PI), ... , (IXn, Pn) in I
whose lengths satisfy
n
L (P; - lXi) < fJ. (2)
;= I
Such anfis obviously continuous: simply take n = 1.
In the following theorem, the implication (b)-+ (c) is probably the most
interesting. That (a) -+ (c) without assuming monotonicity off is the content
of Theorem 7.20.
7.18 Tbeorem Let I = [a, b], let f: 1-+ RI be continuous and nondecreasing.
Each of the following three statements about f implies the other two:
(a) fis AC on I.
(b) fmaps sets of measure 0 to sets of measure O.
(c) fis differentiable a.e. on I,f' E E, and
f(x) - f(a) = IX !'(t) dt (IX :s; X :s; b). (1)
Note that the functions constructed in Example 7.16(b) map certain compact
sets of measure 0 onto the whole unit interval!
Exercise 12 complements this theorem.
PROOF We will show that (a)-+ (b)-+ (c)-+ (a).
Let Wi denote the u-algebra of all Lebesgue measurable subsets of RI.
Assumefis AC on I, pick E c: I so that E E Wi and m(E) :;: O. We have to
show thatf(E) E Wi and m(f(E)) = O. Without loss of generality, assume that
neither a nor b lie in E.
Choose E > O. Associate fJ > 0 to f and E, as in Definition 7.17. There is
then an open set V with m(V) < fJ, so that E c: V c: I. Let (IX;, PJ be the
disjoint segments whose union is V. Then L (P; - IXI) < fJ, and our choice of fJ
shows that therefore
L (f(P;) - f(IX;)) :s; E. (2)
;
[Definition 7.17 was stated in terms of finite sums; thus (2) holds for every
partial sum of the (possibly) infinite series, hence (2) holds also for the sum of
the whole series, as stated.]
Since E c: V,f(E) c: U [f(IX;),f(P;)]. The Lebesgue measure of this union
is the left side of (2). This says that f(E) is a subset of Borel sets of arbitrarily
small measure. Since Lebesgue measure.is complete, it follows thatf(E) E Wi
and m(f(E)) = O.
We have now proved that (a) implies (b).