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

Question

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.

Answer 1

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

1. Some subset of the Cantor set.
2. Some intervals corresponding to the gaps in the Cantor set.

Any set of form (1) is Lebesgue measurable, because the Lebesgue measure is complete: a subset of a measure zero set is measurable.

Any set of form (2) is Lebesgue measurable, because it's an at most countable union of intervals.