1549901369-Elements_of_Real_Analysis__Denlinger_

170 Chapter 3 • Topology of the Real Number System

SETS OF MEASURE ZERO

Is the Cantor set a large or small subset of [O, l]? In one sense it is a large
subset. It is uncountable; in Theorem 3.4.12 we showed that it has the same
number of elements as [O, l]. In another sense, it is small in comparison with
[O, l]. As we shall see below, it has total length (measure) zero, whereas [O, 1]
has total length 1. These two views of the Cantor set, as simultaneously large
in one sense and small in another, suggest that it is a peculiar set. Indeed, it
is one of the most peculiar sets in all of mathematics. It is a source of many
intriguing, non-intuitive examples and counterexamples in analysis.

Definition 3.4.19 A set A ofreal numbers has measure zero if Ve> 0, A can
be covered by a countable collection of open intervals of total length less than c.
That is, A has measure zero iff Ve > 0 :3 collection {In : n E N} of open intervals
00
In= (an, bn) such that L length(In) < c, where length(In) = (bn - an)·
n=l

Theorem 3.4.20 A countable set must have measure zero.

Proof. Let A = { Xn : n E N} be a countable set. Let c > 0. Then \In E N,
let

Then Un : n E N} is a countable collection of open intervals that covers A.

(
Moreover, length(In) = Xn + c ) ( c ) 2c c
2 n+z - Xn - 2 n+z = 2 n+z = 2 n+l. Thus,
00 00
L L

c c c c

length(In) = -2n+l = - 4 + - + - + 8 16 · · ·

n=l n=l

~ 4 [1 + 1. 2 + 1. 4 + 1. 8 + ... ] = ~ 4. 2

(sum of geometric series)

c

2 < c.

Thus, A has measure zero. •

Theorem 3.4.21 The Cantor set has measure zero.

Proof. Recall the construction of the Cantor set: