Prove that the distributional derivative of the Cantor function is the the $\log_3 2$-Hausdorff measure restricted to the Cantor set

Question

The Cantor function has weak derivative equal to $0$ a.e.

Its distributional derivative should be the $\log_3 2$-Hausdorff measure restricted to the Cantor set, but I'm having troubles doing the computation because of the peculiar definition of the Cantor function.

How can we compute the distributional derivative of Cantor function?

Answer 1

Use the self-similarity.

You want to show that the characteristic function of [0,a) integrated against the derivative is equal to the Hausdorff measure of $[0,a)\cap C$ (and equal to the Cantor function at a). That is true for a=1 (by suitable normalization), for a=1/3 by self-similarity of the Cantor function and then true for $1/3<a<2/3$ trivially, since the function is constant on that interval. Expand a in base 3 and use the self-similarity to show the general result.