Certain problems in the theory of Choquet simplexes connected with harmonic spaces (Q1070171)

Let S be a metrizable Choquet simplex and E(S) be the set of its boundary points. Let \(\mu_ x\) be the maximal measure which represent \(x\in S\). Let \(B\) be the set of all bounded real-valued Borel functions on E(S); for \(f\in B\) and \(x\in S\) define \(u_ f(x)=\mu_ x(f)\). Let \(A_ f=\{x\in S:\) \(u_ f\) is continuous at \(x\}\). Among other problems, the author studies the properties of the sets \(A_ 0=\cap \{A_ f: f\in B\}\) and \(S\setminus A_ 0\), and elucidates the connection of the set \(Sh_ S=\overline{E(S)}\setminus E(S)\) with the dense parts of S. An example of a nonprimary simplex S that has a dense part \(D\subset A_ 0\) and in which \(Sh_ S\neq \emptyset\) is constructed. Applications in potential theory are given; it is shown that if x is an irregular boundary point of the domain \(\omega \subset R^ n\), then for each neighborhood V(x)\(\subset {\bar \omega}\) there exists a function \(h\in C({\bar \omega})\), harmonic in \(\omega\), such that \(h(x)>0\) and \(h<0\) on \({\bar \omega}\setminus V(x)\).