Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-18T15:15:16.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces

Published online by Cambridge University Press:  28 January 2011

JEAN GOUBAULT-LARRECQ  and
KLAUS KEIMEL
JEAN GOUBAULT-LARRECQ
Affiliation:
LSV, ENS Cachan, CNRS, INRIA Saclay, 61 avenue du Président-Wilson, 94230 Cachan, France Email: goubault@lsv.ens-cachan.fr
KLAUS KEIMEL
Affiliation:
Fachbereich Mathematik, Technische Universität, 64289 Darmstadt, Germany Email: keimel@mathematik.tu-darmstadt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish Choquet–Kendall–Matheron theorems on non-Hausdorff topological spaces. This typical result of random set theory is profitably recast in purely topological terms using intuitions and tools from domain theory. We obtain three variants of the theorem, each one characterising distributions, in the form of continuous valuations, over relevant powerdomains of demonic, angelic and erratic non-determinism, respectively.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Issue 3 , June 2011 , pp. 511 - 561
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science 3, Oxford University Press 1–168.Google Scholar
Alvarez-Manilla, M., Edalat, A. and Saheb-Djahromi, N. (1997) An extension result for continuous valuations. In: Edalat, A., Jung, A., Keimel, K. and Kwiatkowska, M. (eds.) Proceedings of the 3rd Workshop on Computation and Approximation (Comprox III). Electronic Notes in Theoretical Computer Science 13.Google Scholar
Birkhoff, G. (1940) Lattice Theory, American Mathematical Society.Google Scholar
Choquet, G. (1953–54) Theory of capacities. Annales de l'Institut Fourier 5 131–295.Google Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral, Kluwer.CrossRefGoogle Scholar
Erné, M. (1991) The ABC of order and topology. In: Herrlich, H. and Porst, H.-E. (eds.) Category Theory at Work. Research and Exposition in Mathematics 18, Heldermann Verlag 57–83.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications 93, Cambridge University Press.Google Scholar
Gilboa, I. and Schmeidler, D. (1992) Additive representation of non-additive measures and the Choquet integral. Discussion Papers 985, Northwestern University, Center for Mathematical Studies in Economics and Management Science.Google Scholar
Goubault-Larrecq, J. (2007a) Continuous capacities on continuous state spaces. In: Arge, L., Cachin, Ch., Jurdziński, T. and Tarlecki, A. (eds.) Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP'07), Wrocław, Poland. Springer-Verlag Lecture Notes in Computer Science 4596 764–776Google Scholar
Goubault-Larrecq, J. (2007b) Continuous previsions. In: Duparc, J. and Henzinger, T. A. (eds.) Proceedings of the 16th Annual EACSL Conference on Computer Science Logic (CSL'07), Lausanne, Switzerland. Springer-Verlag Lecture Notes in Computer Science 4646 542–557.Google Scholar
Goubault-Larrecq, J. (2007) Une introduction aux capacités, aux jeux et aux prévisions, Version 6. Available at http://www.lsv.ens-cachan.fr/~goubault/ProNobis/pp_1_6.pdf.Google Scholar
Goubault-Larrecq, J. (2010) De Groot duality and models of choice: angels, demons and nature. Mathematical Structures in Computer Science 20 (2)169–237.CrossRefGoogle Scholar
Groemer, H. (1978) On the extension of additive functionals on classes of convex sets. Pacific Journal of Mathematics 75 397–410.Google Scholar
Heckmann, R. (1997) Abstract valuations: A novel representation of Plotkin power domain and Vietoris hyperspace. In: Proc. 13th Intl. Symp. on Mathematical Foundations of Programming Semantics (MFPS'97). Electronic Notes in Theoretical Computer Science 6.CrossRefGoogle Scholar
Jones, C. and Plotkin, G. D. (1989) A probabilistic powerdomain of evaluations. In: Proc. 4th IEEE Symposium on Logics in Computer Science (LICS'89), IEEE Computer Society Press 186–195.Google Scholar
Jung, A. (2004) Stably compact spaces and the probabilistic powerspace construction. In: Desharnais, J. and Panangaden, P. (eds.) Domain-theoretic Methods in Probabilistic Processes. Electronic Lecture Notes in Computer Science 87.Google Scholar
Keimel, K. and Lawson, J. (2005) Measure extension theorems for T 0-spaces. Topology and its Applications 149 (1-3)57–83.Google Scholar
Klain, D. A. and Rota, G.-C. (1997) Introduction to Geometric Probability, Lezioni Lincee, Cambridge University Press.Google Scholar
König, H. (1997) Measure and Integration: An Advanced Course in Basic Procedures and Applications, Springer Verlag.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry, Wiley.Google Scholar
Mislove, M. (1998) Topology, domain theory and theoretical computer science. Topology and Its Applications 89 3–59.CrossRefGoogle Scholar
Mislove, M. (2000) Nondeterminism and probabilistic choice: Obeying the law. In: Proc. 11th Conf. Concurrency Theory (CONCUR'00). Springer-Verlag Lecture Notes in Computer Science 1877 350–364.CrossRefGoogle Scholar
Molchanov, I. (2005) Theory of Random Sets, Probability and Its Applications, Springer Verlag.Google Scholar
Norberg, T. (1989) Existence theorems for measures on continuous posets, with applications to random set theory. Mathematica Scandinavica 64 15–51.Google Scholar
Schalk, A. (1993) Algebras for Generalized Power Constructions, Ph.D. thesis, Technische Universität Darmstadt.Google Scholar
Tix, R. (1995) Stetige Bewertungen auf topologischen Räumen, Diplomarbeit, TH Darmstadt.Google Scholar
Tix, R., Keimel, K. and Plotkin, G. (2005) Semantic domains for combining probability and non-determinism. Electronic Notes in Theoretical Computer Science 129 1–104.CrossRefGoogle Scholar