Cantor set

Cantor set

The

Cantor set

C

is the

canonical

example of an

uncountable set

of

measure zero

. We construct

C

as follows.

Begin with the unit interval

C

0

=

[

0

,

1

]

, and remove the open segment

R

1

:=

(

1

3

,

2

3

)

from the middle. We
define

C

1

as the two remaining pieces

C

1

:=

C

0

∖

R

1

=

[

0

,

1

3

]

⁢

⋃

[

2

3

,

1

]

(1)

Now repeat the process on each remaining segment, removing the open set

R

2

:=

(

1

9

,

2

9

)

⁢

⋃

(

7

9

,

8

9

)

(2)

to form the four-piece set

C

2

:=

C

1

∖

R

2

=

[

0

,

1

9

]

⁢

⋃

[

2

9

,

1

3

]

⁢

⋃

[

2

3

,

7

9

]

⁢

⋃

[

8

9

,

1

]

(3)

Continue the process, forming

C

3

,

C

4

,

…

Note that

C

k

has

2

k

pieces.

Also note that at each step, the

endpoints

of each closed segment will stay in the set forever—e.g., the point

2

3

isn’t
touched as we remove sets.

The

Cantor set

is defined as

C

:=

⋂

k

=

1

∞

C

k

=

C

0

∖

⋃

n

=

1

∞

R

n

(4)

Cardinality of the Cantor set

To establish

cardinality

, we want a

bijection

between the Cantor set
and some set whose cardinality we know .

Start at

C

1

, which has two pieces. Mark the left-hand segment “0” and the right-hand segment “1”. Then continue to

C

2

,
and consider only the leftmost pair. Again, mark the segments “0” and “1”, and do the same for the rightmost pair.

Keep doing this all the way down the

C

k

, starting at the left side and marking the segments 0, 1, 0, 1, 0, 1 as you encounter them,
until you’ve labeled the entire Cantor set.

Now, pick a path through the tree starting at

C

0

and going left-left-right-left…and so on. Mark a

decimal point

for

C

0

, and
record the zeros and ones as you proceed. Each path has a unique number based on your decision at each step.
Every point in the Cantor set will have a unique address dependent solely on the pattern of lefts and rights, 0’s and 1’s, required to
reach it.
The Cantor set therefore has the same cardinality as the set of

sequences

of 0’s and 1’s, which is

2

ℵ

0

, the cardinality of the

continuum

.

The Cantor set and ternary expansions

Return, for a moment, to the earlier observation that numbers such as

1

3

,

2

9

, the endpoints of deleted intervals,
are themselves never deleted. In particluar, consider the first deleted interval: the ternary expansions of its constituent numbers are
precisely those that begin

0.1

, and proceed thence with at least one non-zero “ternary” digit further along. Note
also that the point

1

3

, with ternary expansion

0.1

, may also be written

0.0

⁢

2

˙

(or

0.0

⁢

2

¯

), which has no ternary digit

1

.

Similar

descriptions apply to further deleted intervals.
The result is that the Cantor set is precisely those numbers in the set

[

0

,

1

]

that have a ternary expansion contains no digits

1

.

Measure of the Cantor set

Let

μ

be

Lebesgue measure

. The

measure

of the sets

R

k

that we remove during the construction of the Cantor set are

μ

⁢

(

R

1

)

=

2

3

-

1

3

=

1

3

(5)

μ

⁢

(

R

2

)

=

(

2

9

-

1

9

)

+

(

8

9

-

7

9

)

=

2

9

(6)

⁢

⋮

(7)

μ

⁢

(

R

k

)

=

∑

n

=

1

k

2

n

-

1

3

n

(8)

Note that the

R

’s are

disjoint

, which will allow us to sum their measures without worry. In the limit

k

→

∞

, this gives us

μ

⁢

(

⋃

n

=

1

∞

R

n

)

=

∑

n

=

1

∞

2

n

-

1

3

n

=

1

.

(9)

But we have

μ

⁢

(

C

0

)

=

1

as well, so this means

μ

⁢

(

C

)

=

μ

⁢

(

C

0

∖

⋃

n

=

1

∞

R

n

)

=

μ

⁢

(

C

0

)

-

∑

n

=

1

∞

1

2

n

=

1

-

1

=

0

.

(10)

Thus we have seen that the measure of

C

is zero (though see below for more on this topic). How many points are there in

C

?
Lots, as we shall see.

So we have a set of measure zero (very tiny) with uncountably many points (very big). This non-intuitive result is what
makes Cantor sets so interesting.

Cantor sets with positive measure

Clearly, Cantor sets can be constructed for all sorts of “removals”—we can remove middle halves, or thirds, or any amount

1

r

,

r

>

1

we like. All of these Cantor sets have measure zero, since at each step

n

we end up with

L

n

=

(

1

-

1

r

)

n

(11)

of what we started with, and

lim

n

→

∞

⁡

L

n

=

0

for any

r

>

1

.

However, it is possible to construct Cantor sets with

positive
measure

as well; the key is to remove less and less as we proceed. These Cantor sets have the same “shape” (

topology

) as the
Cantor set we first constructed, and the same cardinality, but a different “size.”

Again, start with the unit interval for

C

0

, and choose a number

0

<

p

<

1

. Let

R

1

:=

(

2

-

p

4

,

2

+

p

4

)

(12)

which has length (measure)

p

2

. Again, define

C

1

:=

C

0

∖

R

1

. Now define

R

2

:=

(

2

-

p

16

,

2

+

p

16

)

⁢

⋃

(

14

-

p

16

,

14

+

p

16

)

(13)

which has measure

p

4

. Continue as before, such that each

R

k

has measure

p

2

k

; note again that all the

R

k

are disjoint. The resulting Cantor set has measure

μ

⁢

(

C

0

∖

⋃

n

=

1

∞

R

n

)

=

1

-

∑

n

=

1

∞

μ

⁢

(

R

n

)

=

1

-

∑

n

=

1

∞

p

⁢

2

-

n

=

1

-

p

>

0

.

Thus we have a whole family of Cantor sets of positive measure to accompany their vanishing brethren.

Title

Cantor set

Canonical name

CantorSet

Date of creation

2013-03-22 12:21:37

Last modified on

2013-03-22 12:21:37

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

34

Author

yark (2760)

Entry type

Definition

Classification

msc 28A80

Related topic

Countable

Related topic

Measure

Related topic

SingularFunction

Related topic

CantorFunction

Related topic

MengerSponge

Related topic

RepresentationTheoremForCompactMetricSpaces

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