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.
