Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T14:12:09.738Z Has data issue: false hasContentIssue false
  • English
  • Français

Measure Convex and Measure Extremal Sets

Published online by Cambridge University Press:  20 November 2018

Petr Dostál ,
Jaroslav Lukeš  and
Jiří Spurný
Petr Dostál
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Jaroslav Lukeš
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Jiří Spurný
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

Keywords

46A5552A07measure convex setmeasure extremal setface
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 536 - 548
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Alfsen, E. M., Compact Convex Sets and Boundary Integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete 57, Springer-Verlag, New York, 1971.Google Scholar
[2] Bauer, H., Measure and Integration Theory. de Gruyter Studies in Mathematics 26, Walter de Gruyter, Berlin, 2001.Google Scholar
[3] Choquet, G., Remarque à propos de la démonstration de l’unicité de P. A. Meyer. Séminaire Brelot–Choquet–Deny (Théorie de Potentiel) 1961/62.Google Scholar
[4] Fremlin, D. H. and Pryce, J. D., Semi-extremal sets and measure representations. Proc. London Math. Soc. 29(1974), 502–520.Google Scholar
[5] Holický, P. and Pelant, J., Internal descriptions of absolute Borel classes. Topology Appl. 141(2004), 87–104.Google Scholar
[6] Holický, P. and Spurný, J., Perfect images of absolute Souslin and absolute Borel Tychonoff spaces. Topology Appl. 131(2003), no. 3, 281–294.Google Scholar
[7] Odell, E. and Rosenthal, H. P., A double-dual characterization of separable Banach spaces containing l1 . Israel J. Math. 20(1975), no. 3–4, 375–384.Google Scholar
[8] Phelps, R. R., Lectures on Choquet's Theorem. Van Nostrand, Princeton, NJ, 1966.Google Scholar
[9] Raja, M., On some class of Borel measurable maps and absolute Borel topological spaces. Topology Appl. 123(2002), no. 2, 267–282.Google Scholar
[10] von Weizsäcker, H., A note on infinite dimensional convex sets. Math. Scand. 38(1976), no. 2, 321–324.Google Scholar
[11] Winkler, G., Choquet order and simplices with applications in probabilistic models. Lecture Notes in Mathematics 1145, Springer-Verlag, Berlin, 1985.Google Scholar