Is it possible to "approximate" a compact set of the plane by compact sets with smooth boundary? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:11:21Z http://mathoverflow.net/feeds/question/75600 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75600/is-it-possible-to-approximate-a-compact-set-of-the-plane-by-compact-sets-with-s Is it possible to "approximate" a compact set of the plane by compact sets with smooth boundary? analyst11 2011-09-16T14:25:56Z 2011-09-16T14:25:56Z <p>Hi,</p> <p>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 $\mathbb{C}_{\infty} \setminus K$ </p> <p>and continuous to the boundary of $K$, where $\mathbb{C}_{\infty}$ is the Riemann sphere.</p> <p>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.</p> <p>My question is the following :</p> <p>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 $g \in A(K)$?</p> <p>Thank you</p>