I want to see why holomorphic mappings send sets of measure zero to sets of measure zero.
I found this statement reading about quasiconformal mappings. In fact, there is a theorem (see for instance L. Ahlfors (2006) Lectures on Quasiconformal Mappings, American Mathematical Society) that state that a quasiconformal mapping $\phi$ maps sets of measure zero to sets of measure zero. I don't know if it could help. It also states that for every measurable set $E$ \begin{equation*} m(\phi(E)) = \int_E \text{Jac}\phi\;dm, \end{equation*} where $\text{Jac}$ denotes the Jacobian and $m$ is the Lebesgue measure.
What I thought is to use the fact that an holomorphic map $f$ is locally quasiconformal almost everywhere (a.e. because of the isolated zeros of $f'$ and the fact that conformality implies locally quasiconformality), but sincerely I'm really stuck.
Thanks in advance!