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.

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 08:13

score 3 · Accepted Answer · answered Feb 23 '18 at 18:15

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

Some subset of the Cantor set.
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.

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

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