Geodesic
On the Dirichlet problem for functions of the first Baire class
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 721-728.

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Let $\Cal H$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname{Ch}X$ of $\Cal H$ is an $F_\sigma$-set if and only if given any bounded Baire-one function $f$ on $\operatorname{Ch}X$ there is an $\Cal H$-affine bounded Baire-one function $h$ on $X$ such that $h=f$ on $\operatorname{Ch}X$. This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$.
Classification : 26A21, 31B05, 31C45, 46A55
Keywords: weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions 
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Spurný, Jiří. On the Dirichlet problem for functions of the first Baire class. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 721-728. http://geodesic.mathdoc.fr/item/CMUC_2001__42_4_a11/