Define the function $g: [0, 1] \rightarrow [0, 1]$ with $$g(y) := \inf\{x \in [0, 1] : f(x) = y\},$$ where $f$ is the Cantor function.
Now let $V \subset [0, 1]$ be a set that is not Lebesgue measurable. Show that $g(V)$ is Lebesgue measurable, but not Borel measurable.
I managed to do answer the first part, but I'm struggling with proving that it's not Borel measurable. Help would be appreciated!