# Cantor Function

## Cantor ternary Function

if ${\displaystyle {\mathcal {C}}}$ is the Cantor set on [0,1], then the Cantor function c : [0,1] → [0,1] can be defined as[1]

${\displaystyle c(x)={\begin{cases}\sum _{n=1}^{\infty }{\frac {a_{n}}{2^{n}}},&x=\sum _{n=1}^{\infty }{\frac {2a_{n}}{3^{n}}}\in {\mathcal {C}}\ \mathrm {for} \ a_{n}\in \{0,1\};\\\sup _{y\leq x,\,y\in {\mathcal {C}}}c(y),&x\in [0,1]\setminus {\mathcal {C}}.\\\end{cases}}}$

### Properties of Cantor Functions

• Cantor Function is continuous everywhere, zero derivative almost everywhere.
• lack of absolute continuity.
• Monotonicity
• Its value goes from 0 to 1 as its argument reaches from 0 to 1.

## Cantor Function Alternative

The Cantor Function can be constructed iteratively using homework construction.[2]

## Measurable function with pre-image of Lebesgue measurable set not Lebesgue measurable

Define ${\displaystyle f(x)={\begin{cases}\sum _{n=1}^{\infty }{\frac {2b_{n}}{3^{n}}},&x=\sum _{n=1}^{\infty }{\frac {b_{n}}{2^{n}}}\ \mathrm {for} \ b_{n}\in \{0,1\}\\0\ \mathrm {otherwise} \\\end{cases}}}$

Then it can be shown ${\displaystyle f(x)}$ is the pointwise limit of simple functions. However f takes values in the cantor set on the set of non terminating decimals. We can find a non measurable set ${\displaystyle F}$ such that ${\displaystyle E:=f(F)}$ is a null set and thus lebesgue measurable. Therefore ${\displaystyle f^{-1}(E)}$ fails to be Lebesgue measurable despite E being measurable.

This is analogous to the construction in HW5 2d) to 2 g) and is useful for motivating the definition of the Lebesgue measurable functions to be ${\displaystyle ({\mathcal {L}},{\mathcal {B}}_{\overline {\mathbb {R} }})}$ measurable

## References

1. Terence Tao, An introduction to measure theory

1. Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). Expositiones Mathematicae. Elsevier BV. 24 (1): 1–37.
2. Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.