Definition. The Cantor-Lebesgue function is defined on by
In this definition, we choose the expansion of in which or .
Theorem. The Cantor-Lebesgue function is not absolutely continuous on .
Proof. Choose . Then for every , there is a cover such that so that since has measure zero.
But the Cantor-Lebesgue function only changes on the Cantor set only. So So it is not absolutely continuous.