Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-5dd2w Total loading time: 0.225 Render date: 2022-05-23T10:20:18.827Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces

Published online by Cambridge University Press:  28 January 2011

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

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
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 1168.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.CrossRefGoogle Scholar
Choquet, G. (1953–54) Theory of capacities. Annales de l'Institut Fourier 5 131295.CrossRefGoogle 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 5783.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 764776CrossRefGoogle 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 542557.CrossRefGoogle 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)169237.CrossRefGoogle Scholar
Groemer, H. (1978) On the extension of additive functionals on classes of convex sets. Pacific Journal of Mathematics 75 397410.CrossRefGoogle 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 186195.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)5783.CrossRefGoogle 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 89 359.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 350364.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 1551.CrossRefGoogle 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 1104.CrossRefGoogle Scholar
9
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *