1549901369-Elements_of_Real_Analysis__Denlinger_

7.9 *Lebesgue's Criterion for Riemann Integrability 451

Hence, \:/c: > 0, Xe can b e squeezed between two step functions, 0 and Xena

such that

(55)
By Theorem 7.4.14, this means that xe is integrable on [O, l]. Moreover, by
inequality (55) and the forcing principle,

l; Xe= 0. D

CONDITIONS HOLDING "ALMOST EVERYWHERE"

Definition 7.9.10 A condition P is said to hold almost everywhere in a
set A if { x E A : x does not satisfy condition P} has measure 0.

The t erm "almost everywhere" is sometimes useful in describing a situation
in analysis. For example, Lebesgue's criterion could be rephrased, "f is Riemann
integrable on [a , b] iff f is continuous almost everywhere in [a, b]." As another
example, recall the Cantor function <p defined in Section 5.5. There we proved
that <pis continuous and monotone increasing on [O, l], <p(O) = 0 and <p(l) = 1.
Yet in Exercise 6.3.13 we saw that <p^1 (x) = 0 almost everywhere in [O, l]. We
shall see further uses of this language in the exercises below.

EXERCISE SET 7.9

1. Prove that
   (a) a subset of a set of measure 0 has measure 0.
   (b) the union of two sets of measure 0 has measure 0.
   (c) the union of finitely many sets of measure 0 has measure 0.
   (d) the union of countably many sets of measure 0 has measure 0.


2. Suppose f(x ) ~ 0 on [a, b], and l: f = 0. Prove that
   (a) \:/c > 0, {x E [a, b]: f(x ) ~ c} has measure 0. (See Exercise 7.2.12.)
   (b) f(x) = 0 almost everywhere on [a, b].


3. Suppose f and g are Riemann integrable on [a, b], and l: If -gJ = 0.
   Prove tha t f(x ) = g(x ) almost everywhere on [a, b].


4. Suppose f is Riem ann integrable on [a, b], and f(x) = g(x) almost ev-
   erywhere on [a, b]. Must g also be integrable on [a, b]? (Compare with
   Theorem 7.4.9.)


5. Suppose f is Riemann integrable on [a, b], and define Fon [a, b] by F(x) =
   l: f. Prove that F'(x ) = f(x) almost everywhere on [a, b].