A note on Mergelyan's theorem (Q792501)

Suppose that X is a compact subset of the complex plane and that the complement of X is connected. Let A(X) denote the set of functions which are continuous on X and analytic on the interior of X. Let R(X) be the subset of A(X) consisting of uniform limits of polynomials. Mergelyan's theorem states that \(R(X)=A(X)\) [\textit{S. N. Mergelyan}, Dokl. Akad. Nauk SSSR, N. Ser. 78, 405-408 (1951; Zbl 0042.082)]. The author gives a direct proof of this theorem based on analysis of the associated spaces of annihilating measures. His proof uses, among other things, ideas of \textit{E. Bishop} [Duke Math. J. 27, 331-340 (1960; Zbl 0094.272)].