The exercise says: Use the fact that #$\mathcal{B}(\mathbb{R}) = $#$\mathbb{R}$ (borel sets) to show that if $f:\mathbb{R} \to \mathbb{R}$ is a bijection such that there is a Lebesgue Measurable set $C$ with $m(C) = 0$ but $m(f^{-1}(C))>0$ then $f$ is not measurable as a map from $(\mathbb{R}, \mathcal{L}) \to (\mathbb{R},\mathcal{L})$ being $\mathcal{L}$ the Lebesgue $\sigma$-algebra.
This excercise misleads me because I would do it in another way! Indeed, I would take a non-measurable set $N \subset f^{-1}(C)$ (I can because $f^{-1}(C)$ has positive measure and hence I can construct a Vitali-like set.), then I consider $X = f(N) \subset C$. Now $X \in \mathcal{L}$ because it has exterior measure 0, but $f^{-1}(X) = N \notin \mathcal{L}$, so we can conclude that $f$ is note measurable as a maps from $(\mathbb{R}, \mathcal{L}) \to (\mathbb{R}, \mathcal{L})$. ¿Is this ok, isn't?
In any case, I would like to understand the idea of the excercise and I stuck on this. I have tried somethings like: "Every subset of $C$ is measurable, so I have $|\mathcal{P(\mathbb{R})}|$ measurable sets inside $C$, but just $|\mathbb{R}|$ Borel sets inside $f^{-1}(C)$..." but I cant get it... Any ideas?