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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.

##### MSC:
 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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