On local compactness of spaces of continuous valuations

Authors

  • Jean Goubault-Larrecq Université Paris-Saclay

Keywords:

Continuous valuations, spaces of measures, local compactness, previsions

Abstract

We show that the spaces of continuous valuations, resp. subprobability valuations on a locally compact space is locally compact; similarly with probability valuations on locally compact, compact spaces. Continuous valuations are close cousins of measures. No separation property is assumed.

References

Wolfgang Adamski, $\tau$-smooth Borel measures on topological spaces. Mathematische Nachrichten, 78:97-107, 1977.

Mauricio Alvarez-Manilla, Achim Jung, and Klaus Keimel, The probabilistic powerdomain for stably compact spaces. Theoretical Computer Science, 328(3):221-244, 2004.

Patrick Billingsley, Probability and Measure. Wiley series in probability and mathematical statistics. John Wiley and Sons, 3rd edition, 1995.

Patrick Billingsley, Convergence of Probability Measures. Wiley series in probability and mathematical statistics. John Wiley and Sons, 2nd edition, 1999.

Matthew de Brecht, Quasi-Polish spaces. Annals of Pure and Applied Logic, 164(3):356-381, 2013.

Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, and Zhenchao Lyu, Domain-complete and LCS-complete spaces. Electronic Notes in Theoretical Computer Science, 2019. Presented at the 8th International Symposium of Domain Theory and its Applications (ISDT'19). Available on arXiv:1902.11142 [math.GN].

Abbas Edalat, Domain theory and integration. Theoretical Computer Science, 151:163-193, 1995.

Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie Don Lawson, Michael Mislove, and Dana Stewart Scott, Continuous Lattices and Domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2003.

Jean Goubault-Larrecq. Continuous previsions. In Jacques Duparc and Thomas A. Henzinger, editors, Proceedings of the 16th Annual EACSL Conference on Computer Science Logic (CSL'07), volume 4646 of Lecture Notes in Computer Science, pages 542-557, Lausanne, Switzerland, September 2007. Springer.

Jean Goubault-Larrecq, De Groot duality and models of choice: Angels, demons and nature. Mathematical Structures in Computer Science, 20(2):169-237, 2010.

Jean Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory-Selected Topics in Point-Set Topology, volume 22 of New Mathematical Monographs. Cambridge University Press, 2013.

Jean Goubault-Larrecq, Isomorphism theorems between models of mixed choice. Mathematical Structures in Computer Science, 27(6):1032-1067, September 2017.

Jean Goubault-Larrecq, Products and projective limits of continuous valuations on $T_0$ spaces. Mathematical Structures in Computer Science, 31(2):234-254, 2021.

Jean Goubault-Larrecq, Probabilistic powerdomains and quasi-continuous domains. Topology Proceedings, 60:1-16, 2022.

Reinhold Heckmann, Spaces of valuations. In S. Andima, R. C. Flagg, G. Itzkowitz, Y. Kong, R. Kopperman, and P. Misra, editors, Papers on General Topology and its Applications, volume 806, pages 174-200. Annals of the New York Academy of Science, Dec 1996.

Claire Jones, Probabilistic Non-Determinism. PhD thesis, University of Edinburgh, 1990. Technical Report ECS-LFCS-90-105.

Claire Jones and Gordon Plotkin, A probabilistic powerdomain of evaluations. In Proceedings of the 4 th Annual Symposium on Logic in Computer Science, pages 186-195. IEEE Computer Society Press, 1989.

Klaus Keimel and Jimmie Lawson, $d$-completions and the $d$-topology. Annals of Pure and Applied Logic, 159:292--306, 2009.

Jimmie D. Lawson, Valuations on continuous lattices. In R.-E. Hoffmann, editor, Mathematische Arbeitspapiere, volume 27, pages 204-225, Universität Bremen, 1982.

Yuri Vasilyevich Prokhorov, Convergence of random processes and limit theorems in probability theory. Teoriya Veroyatnostei i ee Primeneniya, 1(2):177-238, 1956. English translation in Theory of Probabilities and its Applications, 1(2), 1956, pages 157-214.

Nait Saheb-Djahromi, Cpo's of measures for nondeterminism. Theoretical Computer Science, 12:19-37, 1980.

Andrea Schalk, Algebras for Generalized Power Constructions. PhD thesis, Technische Hochschule Darmstadt, 1993. Available from http://www.cs.man.ac.uk/~schalk/publ/diss.ps.gz.

Michael B. Smyth, Powerdomains and predicate transformers: A topological view. In Josep Diaz, editor, Automata, Languages and Programming, pages 662-675, Berlin, Heidelberg, 1983. Springer Berlin Heidelberg.

Regina Tix, Stetige Bewertungen auf topologischen Räumen. Diplomarbeit, Technische Hochschule Darmstadt, 1995.

Flemming Topsøe, Compactness and tightness in spaces of measures with the topology of weak convergence. Mathematica Scandinavica, 34:187-210, 1974.

Published

2024-11-23

How to Cite

Goubault-Larrecq, J. (2024). On local compactness of spaces of continuous valuations. Topology Proceedings, 65, 135–148. Retrieved from https://www.topologyproceedings.org/index.php/tp/article/view/206

Issue

Section

Uncategorized

Similar Articles

<< < 12 13 14 15 16 17 18 > >> 

You may also start an advanced similarity search for this article.