We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.
@article{bwmeta1.element.doi-10_2478_s11533-010-0075-7,
author = {Miroslav Ka\v cena and Ji\v r\'\i\ Spurn\'y},
title = {Affine Baire functions on Choquet simplices},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {127-138},
zbl = {1219.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0075-7}
}
Miroslav Kačena; Jiří Spurný. Affine Baire functions on Choquet simplices. Open Mathematics, Tome 9 (2011) pp. 127-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0075-7/
[1] Alfsen E.M., Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb., 57, Springer, New York-Heidelberg, 1971 | Zbl 0209.42601
[2] Argyros S.A., Godefroy G., Rosenthal H.P., Descriptive set theory and Banach spaces, In: Handbook of the Geometry of Banach Spaces, 2, North-Holland, Amsterdam, 2003, 1007–1069 http://dx.doi.org/10.1016/S1874-5849(03)80030-X | Zbl 1121.46008
[3] Armitage D.H., Gardiner S.J., Classical Potential Theory, Springer Monogr. Math., Springer, London, 2001 | Zbl 0972.31001
[4] Asimow L., Ellis A.J., Convexity Theory and its Applications in Functional Analysis, London Math. Soc. Monogr., 16, Academic Press, London-New York, 1980 | Zbl 0453.46013
[5] Bauer H., Šilovscher Rand und Dirichletsches Problem, Ann. Inst. Fourier Grenoble, 1961, 11, 89–136 | Zbl 0098.06902
[6] Bauer H., Simplicial function spaces and simplexes, Exposition. Math., 1985, 3(2), 165–168 | Zbl 0564.46007
[7] Bliedtner J., Hansen W., Simplicial characterization of elliptic harmonic spaces, Math. Ann., 1976, 222(3), 261–274 http://dx.doi.org/10.1007/BF01362583 | Zbl 0314.31010
[8] Bliedtner J., Hansen W., Potential Theory, Universitext, Springer, Berlin, 1986
[9] Boboc N., Cornea A., Convex cones of lower semicontinuous functions on compact spaces, Rev. Roumaine Math. Pures Appl., 1967, 12, 471–525 | Zbl 0155.17301
[10] Capon M., Sur les fonctions qui verifient le calcul barycentrique, Proc. Lond. Math. Soc., 1976, 32(1), 163–180 http://dx.doi.org/10.1112/plms/s3-32.1.163 | Zbl 0313.46003
[11] Choquet G., Lectures on Analysis I–III, W.A. Benjamin, New York-Amsterdam, 1969 | Zbl 0181.39602
[12] Fremlin D.H., Measure Theory, vol. 4, Torres Fremlin, Colchester, 2003
[13] Johnson W.B., Lindenstrauss J., Basic concepts in the geometry of Banach spaces, In: Handbook of the geometry of Banach spaces, 1, North-Holland, Amsterdam, 2001, 1–84 http://dx.doi.org/10.1016/S1874-5849(01)80003-6 | Zbl 1011.46009
[14] Kačena M., Products and projective limits of function spaces, Comment. Math. Univ. Carolin., 2008, 49(4), 547–578 | Zbl 1212.46016
[15] Kechris A.S., Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer, New York, 1995 | Zbl 0819.04002
[16] Lacey H.E., The Isometric Theory of Classical Banach Spaces, Grundlehren Math. Wiss., 208, Springer, Berlin-Heidelberg-New York, 1974 | Zbl 0285.46024
[17] Lukeš J., Malý J., Netuka I., Smrčka M., Spurný J., On approximation of affine Baire-one functions, Israel J. Math., 2003, 134(1), 255–287 http://dx.doi.org/10.1007/BF02787408 | Zbl 1031.35011
[18] Phelps R.R., Lectures on Choquet's Theorem, Van Nostrand, Princeton-Toronto-New York-London, 1966 | Zbl 0135.36203
[19] Spurný J., Affine Baire-one functions on Choquet simplexes, Bull. Aust. Math. Soc., 2005, 71(2), 235–258 http://dx.doi.org/10.1017/S0004972700038211 | Zbl 1075.46005
[20] Spurný J., On the Dirichlet problem of extreme points for non-continuous functions, Israel J. Math., 2009, 173(1), 403–419 http://dx.doi.org/10.1007/s11856-009-0098-6 | Zbl 1190.46009
[21] Spurný J., The Dirichlet problem for Baire-two functions on simplices, Bull. Aust. Math. Soc., 2009, 79(2), 285–297 http://dx.doi.org/10.1017/S0004972708001263 | Zbl 1180.46006
[22] Spurný J., Baire classes of Banach spaces and strongly affine functions, Trans. Amer. Math. Soc., 2010, 362(3), 1659–1680 http://dx.doi.org/10.1090/S0002-9947-09-04841-7 | Zbl 1215.46009
[23] Talagrand M., A new type of affine Borel function, Math. Scand., 1984, 54(2), 183–188 | Zbl 0562.46005