Let $g(x)=f(x)+x$, where $f(x)$ is the Cantor function from $[0,1]$ to $[0,1]$. prove that $B$ is Lebesgue measurable but not Borel measurable.

Question

Let $g(x)=f(x)+x$, where $f(x)$ is the Cantor function from $[0,1]$ to $[0,1]$. We know for the Cantor set $C$, $g(C)$ contains a nonmeasurable set A. Let $B=g^{-1}(A)$, prove that $B$ is Lebesgue measurable but not Borel measurable.

I can show the first one since $B$ is the subset of a null set, which means $B$ is measurable. What about the second one?