Jensen Measures for R (K )

Abstract

Let K be a compact subset of the complex plane, let R(K) be the uniform algebra on K generated by the rational functions with poles off K, and let H(K) be the space of uniform limits on K of functions harmonic in neighborhoods of K. A decomposition theorem for measures in H(K)⊥ is obtained and used to study the sets of Jensen measures and Arens-Singer measures for points of K. It is proved that if p ɛ K is not a Jensen boundary point for R(K), then the probability measures in the smallest weak-star closed affine space containing the Jensen measures are precisely the Arens-Singer measures for p that are supported on the closure of the component of the fine interior of K containing p. In most cases, though not in all cases, this includes all Arens-Singer measures for p.