Math4121 Lecture 19

Continue on the “small set”

Cantor set

Theorem: Cantor set is perfect, nowhere dense

Proved last lecture.

Other construction of the set by removing the middle non-zero interval $(\frac{1}{n},n>0)$ and take the intersection of all such steps is called $SVC(n)$

Back to $\frac{1}{3}$ Cantor set.

Every step we delete $\frac{2^{n-1}}{3^n}$ of the total “content”.

Thus, the total length removed after infinitely many steps is:

\sum_{n=1}^{\infty} \frac{2^{n-1}}{3^n} = \frac{1}{3}\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n=1

However, the quarter cantor set removes $\frac{3^{n-1}}{4^n}$ of the total “content”, and the total length removed after infinitely many steps is:

Every time we remove $\frac{1}{4^n}$ of the remaining intervals. So on each layer, we remove $\frac{2^{n-1}}{4^n}$ of the total “content”.

So the total length removed is:

\begin{aligned} 1-\frac{1}{4}-\frac{2}{4^2}-\frac{2^2}{4^3}-\cdots&=1-\frac{1}{4}\sum_{n=0}^{\infty} \left(\frac{2}{4}\right)^n\\ &=1-\frac{1}{4}\cdot\frac{1}{1-\frac{2}{4}}\\ &=1-\frac{1}{4}\cdot\frac{4}{2}\\ &=1-\frac{1}{2}\\ &=\frac{1}{2} \end{aligned}

Generalized Cantor set (SVC(n))

The outer content of $SVC(n)$ is $\frac{n-3}{n-2}$ .

Monotonicity of outer content

If $S\subseteq T$ , then $c_e(S)\leq c_e(T)$ .

Proof of Monotonicity of outer content

If $C$ is cover of $T$ , then $S\subseteq T\subseteq C$ , so $C$ is a cover of $S$ . Since $c_e(s)$ takes the inf over a larger set that $c_e(T)$ , $c_e(S) \leq c_e(T)$ .

Theorem Osgood’s Lemma

Let $S$ be a closed, bounded set in $\mathbb{R}$ , and $S_1\subseteq S_2\subseteq \ldots$ , and $S=\bigcup_{n=1}^{\infty} S_n$ . Then $\lim_{k\to\infty} c_e(S_k)=c_e(S)$ .