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Jensen Measures for R ( K )
Author(s) -
Gamelin T. W.,
Lyons T. J.
Publication year - 1983
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-27.2.317
Subject(s) - mathematics , closure (psychology) , boundary (topology) , space (punctuation) , combinatorics , point (geometry) , affine plane (incidence geometry) , affine transformation , plane (geometry) , harmonic function , pure mathematics , discrete mathematics , mathematical analysis , geometry , computer science , economics , market economy , operating system
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 .