Jensen measures of uniform algebras
It is proved that Jensen measures are the maximal measures for certain systems of Jensen measures. More precisely: Let A be a uniform algebra with maximal ideal space. The cone of A-subharmonic functions in \(C_ R(Y)\) defines a partial order \(\prec\) among positive measures on \(M^+(Y)\). For each \(p\in Y\), let \(F_ p\) denote the set of compact subsets of Y containing p. A subset \(\{\lambda_ G| G\in F_ p\}\) of \(M^+(Y)\) is called a consistent system if it satisfies the following conditions: (a) \(\lambda_ G\) is a Jensen measure for p with support contained in G, (b) if \(F\subset G\) then \(\lambda_ F\prec \lambda_ G,\) (c) if \(F\in G\) then \(\lambda_ G(g)\leq \lambda_ F(g)\) for all non- negative \(g\in C_ R(Y)\) such that \(g=0\) on \(G\setminus F.\) There is a unique measure \(\lambda_ Y\) which is maximal in \(\{\lambda_ G| G\in F_ p\}\). A terminal measure is a measure which is maximal in some consistent system. The main result of this paper states that every Jensen measure for A is a terminal measure.
scientific article; zbMATH DE number 4069592
scientific article