Measure, Integration & Real Analysis, 2021a

56 Chapter 2 Measures
2.74 Definition Cantor set
The Cantor set C is [0, 1] \ ( ⋃ ∞
n=1 G n ), where G 1 =( 1 3 , 2 3 ) and G n for n > 1 is
the union of the middle-third open intervals in the intervals of [0, 1] \ ( ⋃ n−1
j=1 G j).
One way to envision the Cantor set C is to start with the interval [0, 1] and then
consider the process that removes at each step the middle-third open intervals of all
intervals left from the previous step. At the first step, we remove G 1 =( 1 3 , 2 3 ).
G 1 is shown in red.
After that first step, we have [0, 1] \ G 1 =[0, 1 3 ] ∪ [ 2 3
,1]. Thus we take the
middle-third open intervals of [0, 1 3 ] and [ 2 3
,1]. In other words, we have
G 2 =( 1 9 , 2 9 ) ∪ ( 7 9 , 8 9 ).
G 1 ∪ G 2 is shown in red.
Now [0, 1] \ (G 1 ∪ G 2 )=[0, 1 9 ] ∪ [ 2 9 , 1 3 ] ∪ [ 2 3 , 7 9 ] ∪ [ 8 9
,1]. Thus
G 3 =( 1
27 , 2
27 ) ∪ ( 7 27 , 8 27 ) ∪ ( 19
27 , 20
27 ) ∪ ( 25
27 , 26
27 ).
G 1 ∪ G 2 ∪ G 3 is shown in red.
Base 3 representations provide a useful way to think about the Cantor set. Just
as
10 1 = 0.1 = 0.09999 . . . in the decimal representation, base 3 representations
are not unique for fractions whose denominator is a power of 3. For example,
1
3 = 0.1 3 = 0.02222 . . . 3 , where the subscript 3 denotes a base 3 representations.
Notice that G 1 is the set of numbers in [0, 1] whose base 3 representations have
1 in the first digit after the decimal point (for those numbers that have two base 3
representations, this means both such representations must have 1 in the first digit).
Also, G 1 ∪ G 2 is the set of numbers in [0, 1] whose base 3 representations have 1 in
the first digit or the second digit after the decimal point. And so on. Hence ⋃ ∞
n=1 G n
is the set of numbers in [0, 1] whose base 3 representations have a 1 somewhere.
Thus we have the following description of the Cantor set. In the following
result, the phrase a base 3 representation indicates that if a number has two base 3
representations, then it is in the Cantor set if and only if at least one of them contains
no 1s. For example, both 1 3 (which equals 0.02222 . . . 3 and equals 0.1 3 ) and 2 3 (which
equals 0.2 3 and equals 0.12222 . . . 3 ) are in the Cantor set.
Measure, Integration & Real Analysis, by Sheldon Axler