The support of representing measures for<i>R</i>(<i>X</i>)

Abstract

There are a couple of recent results about algebras of rational functions in the plane with essentially the same method of proof. One result states that the nontrivial Gleason parts of the function algebra R(X) have positive Lebesgue planar measure. A second asserts the lack of completely singular annihilating measures. In this note it is shown how with little extra effort the same method of proof provides even more information about R(X). Specifically it is shown that representing measures for R(X) actually represent for uniform limits of rational functions whose poles lie off the closure of a part. The most noteworthy corollary establishes that the closure of a part must be connected. Also included is a brief summary of the proofs of the two previously known results mentioned above.