3.4 *The Cantor Set 173
2
Then, let R2 = the union of the 2 open intervals of length r 2 , one centered
in each of the 2 closed intervals comprising C 1. [Note that 0 < r < ~ =>
0 < r < 1 - r and 0 < r; <^1 ;r .] Then the set C2 = [O, l] - (R1 U R2)
consists of 4 (= 22 ) disjoint closed intervals. Moreover, μ(R 2 ) = 2 · r; = r^2 and
μ(C2) = 1-(r + r^2 ).
Continuing inductively, we let Rn+ 1 consist of the 2n open intervals of
length r;:
1
, one centered in each of the 2n disjoint closed intervals comprising
Cn. Then
n
We define Cn+I = [O, l] - LJ Ri, and calculate its measure:
i=l
μ(Cn+1) = 1-μ (Q
1
~) = 1-iti ri.
Finally, we define
00
C(a) = n Cn.
n=l
Then by (μ7), since C1 2 C2 2 · · · 2 Cn 2 · · ·,
μ (C(a)) = lim μ(Cn) = lim μ(Cn+I)
= nl: [1 -I>:]-+oo
n-H:x:> i=l
00
= 1 - I: ri = 1 - (1 - a)
i=l
=a.
The set C(a) is a "fat" Cantor-like set of measure a. •
In Exercise 3.4.19 you will explore some of the properties of such "fat" sets.
EXERCISE SET 3.4
l. Prove Theorem 3.4.2.
- Prove Theorem 3.4.3.
- Prove that every point of C is a boundary point of C.
- Prove Lemma 3.4.4.
5. Prove Theorem 3.4.10.
6. (a) Prove that the set of endpoints of all the open intervals removed from
[O, l] in forming C is a countable set.
(b) Prove that there are uncountably many members of C that are not
endpoints of removed intervals.