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On the Dirichlet problem for functions of the first Baire class. (English) Zbl 1090.46500
The article contains an answer to a question raised by F. Jellett [Q. J. Math., Oxf. II. Ser. 36, 71–73 (1985; Zbl 0582.46010)]. The following theorem is proved: Suppose that \(\mathcal H\) is a simplicial function space on a metric compact space \(X\). Then the Choquet boundary \({\text{Ch}}\, X\) of \(\mathcal H\) is an \(F_\sigma \)-set iff given any bounded Baire-one function \(f\) on \({\text{Ch}}\, X\) there exists an \(\mathcal H\)-affine bounded Baire-one function \(h\) on \(X\) such that \(h=f\) on \({\text{Ch}}\, X\). As a consequence, a necessary and sufficient condition for semiregularity of a set is given.

46A55 Convex sets in topological linear spaces; Choquet theory
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
26A21 Classification of real functions; Baire classification of sets and functions
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