Problem: Let $f: A \rightarrow \mathbb{C}$ be a holomorphic function, and $f(A)$ is contained in a measure zero set in $\mathbb{C}$. Prove that $f$ is constant.
Attempt at solution: The statement seems to have some connections with the open mapping theorem, but my $A$ here is not necessarily a connected open set so I cannot directly use it here. I can't seem to find a way to use the measure zero image properly. Any help is appreciated!