Let $$ \tilde{\mathbb C}={\mathbb C}\smallsetminus (-\infty,0] $$ the complex plane without the negative real axis. Let $V$ denote the set of all holomorphic functions $f:\tilde{\mathbb C}\to{ \mathbb C}$ such that $$ f^+(0)=\lim_{\lambda\searrow 0}f(\lambda) $$ exists. If $K\subset \{\mathrm{Re}(z)>0\}$ is a compact subset which is not finite, then a given $f\in V$ is uniquely determined by its restriction to $K$. Therefore it is reasonable to ask:

Does there exists a compact set $K$ and a complex valued Radon measure $\mu$ on $K$ such that $$ f^+(0)=\int_K f(\lambda)\,d\mu(\lambda) $$ holds for every $f\in V$?