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 $f\in V$?



Integral representation of a limit
No. The existence of such a representation would imply that the linear functional $\Phi:V\to \mathbb C$, $f\mapsto f^+(0)$ is a continuous with respect to the topology of uniform convergence on compact subsets of $K$. By Hahn-Banach one could thus extend $\Phi$ to $H(\tilde{\mathbb C})$. This is certainly not true: The functions $f\_n(z)= \frac{1}{z+1/n}$ are in $V$ and converge in $H(\tilde{\mathbb C})$. But $\Phi(f\_n)=n$ does not converge.


