Hi,

Let $K \subseteq \mathbb{C}$ compact. Suppose that $K$ is connected, and that the boundary of $K$ is a simple, closed, piecewise $C^1$ curve. Denote by $A(K)$ the set of all functions holomorphic on $\mathbb{C}_{\infty} \setminus K$

and continuous to the boundary of $K$, where $\mathbb{C}_{\infty}$ is the Riemann sphere.

By the Riemann mapping theorem, it is possible to write $K = \cap_n K_n$, where $(K_n)$ is a decreasing sequence of compact sets such that the boundary of each $K_n$ is an analytic, simple and closed curve.

My question is the following :

Is it possible to find a decreasing sequence of compacts $(K_n)$ such that $K=\cap_n K_n$ and the boundary of each $K_n$ is an analytic simple closed curve, with the additional requirement that $$\int_{\partial K_n} g(z) dz \rightarrow \int_{\partial K} g(z)dz$$ as $n \rightarrow \infty$, for each function $g \in A(K)$?

Thank you