Cantor Set

Definition

$\mathcal{C}:=\underset{n\to \infty }{\lim }{C}_n={\bigcap }_{n=0}^{\infty }{C}_n$ where $C_0:=[0,1]$ and $C_n:=\frac{C_{n-1}}{3}\cup \left(\frac{2}{3}+\frac{C_{n-1}}{3}\right)$

Cantor set is created by iteratively deleting the open middle third from a set of line segments.

Facts

The elements of $C$ are numbers consisting of 0 or 2 in ternary

$|\mathcal{C}|=|[0,1]|=2^{\aleph }$ The Cardinality of $\mathcal{C}$ is equal to the interval $[0,1]$

$\lambda (\mathcal{C})=0$ $\mathcal{C}$ is a Lesbesgue-measurable set, and its Lebesgue Measure is $0$