Cantor (-) ternary sets

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Let be a family of intervals defined as follows:

and

with ,

i.e. from every interval of -th generation remove an open interval of length

centred at the mid-point of to get two new (closed) intervals of next generation.

Let

.

Picture for

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.⁰____________________. . ___________________.¹

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The sets are compact (because closed and bdd) and we have

.

Thus (by intersection of compact sets thm) the set

is a compact set.

If , clearly it contains a sequence of points

(which are the end pts of the intervals ).

In fact it contains uncountably many pts (see later).

Note that for any the set is nowhere dense, which means for any open

interval , the intersection is not dense in .

(It is easy to see this from the construction of : For some there is and

we remove from it an open interval.)

Remark: The nowhere dense sets and their countable unions are called the sets

of the 1^st category.

One can characterize the Cantor ternary set ,, as a set of numbers of the form

with . (To avoid nonuniqueness we discard representation for which

for some .) Using this representation we see that

Theorem

The Cantor ternary set is uncountable.

Proof: It is sufficient to show that there is function which is onto.

For this we note that every real number can be represented as follows:

with t_n = 0 or 1. (This representation may be not unique.)

The desired function is obtained by setting

with

Proposition

The total length of removed intervals equals .

Proof: The number of intervals with generations equals . We remove from each of them an interval of length . Thus sum of the lengths of all removed intervals equals

.

Remark In particular for , the sum of lengths of all removed intervals equals ,

i.e. it is equal to the length of the entire interval .

The sets are Lebesgue measurable and

.

In particular

(similarly as for , although is uncountable)!