# Showing that preimage of a subset of $[0,1]$ is Lebesgue measurable under the Cantor function.

Let $C$ be the Cantor function. I am asked to show that for any $A \subset [0,1]$, $C^{-1}(A)$ is Lebesgue measurable.

I've shown so far that the Cantor function is uniformly continuous, increasing and that the image of the cantor set under the cantor function is $[0,1]$.

I don't really know how to start working on this problem so any help would be appreciated.

• What is the set $A$? – user251257 Feb 22 '18 at 19:53
• Any subset of $[0,1]$. – McNuggets666 Feb 22 '18 at 23:36
• For $a\\in[0,1]$, if $a$ have a finite binary expansion, what is $C^{-1}(a)$? If $a$ doesn't haven't a finite expansion, where do $C^{-1}(a)$ belong? – user251257 Feb 23 '18 at 8:13
• Check Proposition 2.5 – Gono Feb 23 '18 at 8:18

For any set $A\subset [0, 1]$, the preimage $C^{-1}(A)$ is the union of: