1549901369-Elements_of_Real_Analysis__Denlinger_

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.

2. Prove Theorem 3.4.3.


3. Prove that every point of C is a boundary point of C.


4. 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.