Measure, Integration & Real Analysis, 2021a

58 Chapter 2 Measures
Now we can define an amazing function.
2.77 Definition Cantor function
The Cantor function Λ : [0, 1] → [0, 1] is defined by converting base 3 representations
into base 2 representations as follows:
• If x ∈ C, then Λ(x) is computed from the unique base 3 representation of
x containing only 0s and2s by replacing each 2 by 1 and interpreting the
resulting string as a base 2 number.
• If x ∈ [0, 1] \ C, then Λ(x) is computed from a base 3 representation of x
by truncating after the first 1, replacing each 2 before the first 1 by 1, and
interpreting the resulting string as a base 2 number.
2.78 Example values of the Cantor function
• Λ(0.0202 3 )=0.0101 2 ; in other words, Λ ( )
20
81 =
5
• Λ(0.220121 3 )=0.1101 2 ; in other words Λ ( 664
729
16
.
) =
13
16 .
• Suppose x ∈ ( 1
3
, 2 )
3 . Then x /∈ C because x was removed in the first step of
the definition of the Cantor set. Each base 3 representation of x begins with 0.1.
Thus we truncate and interpret 0.1 as a base 2 number, getting 1 2
. Hence the
Cantor function Λ has the constant value 1 2 on the interval ( 1
3
, 2 )
3 , as shown on
the graph below.
• Suppose x ∈ ( 7
9
, 8 )
9 . Then x /∈ C because x was removed in the second step
of the definition of the Cantor set. Each base 3 representation of x begins with
0.21. Thus we truncate, replace the 2 by 1, and interpret 0.11 as a base 2 number,
getting 3 4 . Hence the Cantor function Λ has the constant value 3 4
on the interval
( 79
, 8 )
9 , as shown on the graph below.
Graph of the Cantor function on the intervals from first three steps.
Measure, Integration & Real Analysis, by Sheldon Axler