Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$.

My question is how to characterize all such (Radon) measures $\mu$ on $G$, that $\int_{G} f d\mu=0$ for any $f\in H\left(G\right)$.

A little bit on the motivation. I am interested in some internal description of the dual to $H\left(G\right)$. The only description which I know at this moment is the one from Lueking/Rubel. There is only one-dimensional case considered and in this case the dual is shown to be isomorphic to the space of holomorphic functions on $\mathbb{C}^{d}\backslash G$. Since this description relies on the embedding of $G$ into $\mathbb{C}^{d}$, it is not internal.

Possibly the problems are not well-stated, but I believe that they are meaningful.

Thank you.

2more comments