I came across the following problem in my self-study:
Let $f:[0,1] \rightarrow [0,1]$ be the Cantor (ternary) function, and let $g(x) = f(x) + x$. Then:
(a) $g$ is bijection from $[0,1]$ to $[0,2]$, and $h = g^{-1}$ is continuous.
(b) If $C$ is the Cantor (ternary) set, $m(g(C))=1$.
(c) It follows that $g(C)$ contains a Lebesgue nonmeasurable set $A$. Let $B = g^{-1}(A)$, and show that $B$ is Lebesgue measurable but not Borel.
(d) There exist a Lebesgue measurable function $F$ and a continuous function $G$ on $\mathbb{R}$ such that $F \circ G$ is not Lebesgue measurable.
I think I can prove (a), but I am not having any nice ideas on how to proceed for the remaining (3) parts, and I am interested to see if anyone visiting knows how to tackle this interesting exercise.