On the Dirichlet problem for functions of the first Baire class

Jiří Spurný

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 721-728
  • ISSN: 0010-2628

Abstract

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Let be a simplicial function space on a metric compact space . Then the Choquet boundary of is an -set if and only if given any bounded Baire-one function on there is an -affine bounded Baire-one function on such that on . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set .

How to cite

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Spurný, Jiří. "On the Dirichlet problem for functions of the first Baire class." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 721-728. <http://eudml.org/doc/248820>.

@article{Spurný2001,
abstract = {Let $\mathcal \{H\}$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname\{Ch\}X$ of $\mathcal \{H\}$ is an $F_\sigma $-set if and only if given any bounded Baire-one function $f$ on $\operatorname\{Ch\}X$ there is an $\mathcal \{H\}$-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 $X$.},
author = {Spurný, Jiří},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions; weak Dirichlet problem; function space; Choquet simplex; Baire-one function},
language = {eng},
number = {4},
pages = {721-728},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the Dirichlet problem for functions of the first Baire class},
url = {http://eudml.org/doc/248820},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Spurný, Jiří
TI - On the Dirichlet problem for functions of the first Baire class
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 721
EP - 728
AB - Let $\mathcal {H}$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname{Ch}X$ of $\mathcal {H}$ is an $F_\sigma $-set if and only if given any bounded Baire-one function $f$ on $\operatorname{Ch}X$ there is an $\mathcal {H}$-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 $X$.
LA - eng
KW - weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions; weak Dirichlet problem; function space; Choquet simplex; Baire-one function
UR - http://eudml.org/doc/248820
ER -

References

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  8. Jellett F., On affine extensions of continuous functions defined on the extreme boundary of a Choquet simplex, Quart. J. Math. Oxford (2) 36 (1985), 71-73. (1985) Zbl0582.46010MR0780351
  9. Lacey H.E.. Morris P.D., On spaces of type and their duals, Proc. Amer. Math. Soc. 23 (1969), 151-157. (1969) MR0625855
  10. Lukeš J., Malý J., Zajíček L., Fine topology methods in real analysis and potential theory, Lecture Notes in Math. 1189 Springer-Verlag (1986). (1986) MR0861411
  11. Phelps R.R., Lectures on Choquet's theorem, D. Van Nostrand Co. (1966). (1966) Zbl0135.36203MR0193470

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