Category Archives: Measure Theory

The Cantor Set

This is a post regarding the basic properties of the Cantor set.

We start with the definition: The Cantor set is obtained by first constructing a sequence (C_n) of closed sets and then taking the intersection of the sets in this sequence. The sequence is constructed as follows:

1. Start with [0,1] and remove the middle third, i.e. the interval (\frac{1}{3},\frac{2}{3}) to obtain the set C_1=[0,\frac{1}{3}]\cup[\frac{2}{3},1]. So C_1 contains 2 disjoint closed intervals each of length \frac{1}{3}.

2. Next remove the middle third of each of these two intervals leaving C_2=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1] consisting of 2^2 disjoint closed intervals each of length \frac{1}{3^2}.

3. Assuming C_n has been constructed and consists of 2^n disjoint closed intervals each of length \frac{1}{3^n}, remove the middle thirds of all these intervals to obtain C_{n+1} consisting of 2^{n+1} disjoint closed intervals each of length \frac{1}{3^{n+1}}.

By induction we have our sequence (C_n).

Now, we define the Cantor set C as C=\cap_{n=1}^\infty C_n.

We now describe the properties of the Cantor set:

1. In the usual topology on \mathbb{R} the Cantor set is closed (being an intersection of closed sets). Moreover since it is bounded so by the Heine Borel theorem it is compact. What’s extraordinary is that every compact metric space is a continuous image of the Cantor set! The proof may be found here.

2. The Cantor set is uncountable. We prove this by considering the ternary expansion of all numbers in [0,1]. All such numbers may have the form 0.a_1a_2\cdots with a_i\in\{0,1,2\} (including 1=0.22\cdots). We claim that x\in C iff x has a ternary expansion of the form 0.a_1a_2\cdots with a_i\in\{0,2\}.

To prove this consider the construction of C through the ternary lens: In ternary the construction of C_1 involves removing all numbers in (0.011\cdots,0.2) from [0.00\cdots,0.22\cdots]. This shows that in C_1 every number has a ternary expansion of the form 0.a_1a_2\cdots with a_1\in\{0,2\} and a_i\in\{0,1,2\}\forall i>1. Conversely every number of this form is in C_1. Likewise the construction of C_2 involves removing all numbers in (0.0011\cdots,0.02) and in (0.2011\cdots,0.22) from C_1. This shows that in C_2 every number has a ternary expansion of the form 0.a_1a_2\cdots with a_1,a_2\in\{0,2\} and a_i\in\{0,1,2\}\forall i>2. Again conversely every number of this form is in C_2. Continuing in this way we can conclude that x\in C_n iff x=0.a_1\cdots a_na_{n+1}a_{n+2}\cdots with a_1\cdots a_n\in\{0,2\} and a_i\in\{0,1,2\}\forall i>n. Hence by definition of C our claim is established.

Now assume that C is countable and has been enumerated in the list described below:

0.d_1^{(1)}d_2^{(1)}d_3^{(1)}\cdots
0.d_1^{(2)}d_2^{(2)}d_3^{(2)}\cdots
0.d_1^{(3)}d_2^{(3)}d_3^{(3)}\cdots
\cdots

Here each d_i^{(j)}\in \{0,2\}.

Now let x=0.a_1a_2a_3\cdots where a_i=0 if d_i^{(i)}=2 and a_i=2 otherwise. Then x\notin C despite being of the requisite ternary form. Hence C is uncountable.

3. The Cantor set has Lebesgue measure zero. We define a set A to be of Lebesgue measure zero if \forall \epsilon>0\exists a sequence of intervals (I_n) such that A\subset\cup_{n=1}^\infty I_n and \sum_{n=1}^\infty l(I_n)<\epsilon. Here if the end points of I_n are a_n and b_n (with a_n\le b_n) then l(I_n)=b_n-a_n. (The proof that this definition is consistent with the Lebesgue measure may be found in this book.) Now given \epsilon>0 choose n so that (\frac{2}{3})^n<\epsilon. Since C\subset C_n and the total length of C_n is (\frac{2}{3})^n, so it is clear that C is a set of Lebesgue measure zero. Together with point 2, we see that this establishes the existence of uncountable sets of zero measure!

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