Cantor-Lebesgue function

By | October 20, 2015

[업데이트 중]

Definition. The Cantor-Lebesgue function is defined on \mathcal{C} by

    \[ F\left(x\right)=\sum_{k=1}^{\infty}\frac{b_{k}}{2^{k}}\quad\text{if }x=\sum_{k=1}^{\infty}a_{k}3^{-k},\quad\text{where }b_{k}=\frac{a_{k}}{2}. \]

In this definition, we choose the expansion of x in which a_{k}=0 or 2.

Theorem. The Cantor-Lebesgue function is not absolutely continuous on [0,1].

Proof. Choose 0<\varepsilon_0\leq 1. Then for every \delta >0, there is a cover \{(a_k,b_k)\} such that \mathcal{C}\subset\bigcup_{k=1}^\infty (a_k,b_k) so that \sum_{k=1}^{\infty} (b_k -a_k)<\delta since \mathcal{C} has measure zero.

But the Cantor-Lebesgue function F only changes on the Cantor set only. So \sum_{k=1}^{\infty} |f(b_k)-f(a_k)|=1. So it is not absolutely continuous.

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