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Products and projective limits of continuous valuations on T0 spaces

Published online by Cambridge University Press:  14 July 2021

Jean Goubault-Larrecq Open the ORCID record for Jean Goubault-Larrecq [Opens in a new window]
Jean Goubault-Larrecq*
Affiliation:
Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, Gif-sur-Yvette 91190, France
*
Email: goubault@ens-paris-saclay.fr
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Abstract

We show analogues of the Daniell–Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on T0 topological spaces.

Keywords

Projective limitcontinuous valuationmeasureDaniell-Kolmogorov theoremProhorov’s theorem
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 2 , February 2021 , pp. 234 - 254
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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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, vol. 3, Oxford University Press, 1–168.Google Scholar
Alvarez-Manilla, M. (2000). Measure Theoretic Results for Continuous Valuations on Partially Ordered Structures. Phd thesis, Imperial College, London.Google Scholar
Bochner, S. (1955). Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles.CrossRefGoogle Scholar
Daniell, P. J. (1919). Functions of limited variation in an infinite number of dimensions. Annals of Mathematics, Second Series 21 (1) 30–38.CrossRefGoogle Scholar
de Brecht, M. (2013). Quasi-Polish spaces. Annals of Pure and Applied Logic 164 (3) 356–381.CrossRefGoogle Scholar
de Brecht, M., Goubault-Larrecq, J., Jia, X. and Lyu, Z. (2019). Domain-complete and LCS-complete spaces. Electronic Notes in Theoretical Computer Science. Presented at the 8th International Symposium of Domain Theory and its Applications (ISDT’19). Available on arXiv:1902.11142.Google Scholar
Dolecki, S., Greco, G. H. and Lechicki, A. (1995). When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Transactions of the American Mathematical Society 347 (8) 2869–2884.CrossRefGoogle Scholar
Fedorchuk, V. V. (1981). Covariant functors in the category of compacta, absolute retracts, and Q-manifolds. Russian Mathematical Surveys 36 211–233.CrossRefGoogle Scholar
Fujiwara, K. and Kato, F. (2017). Foundations of rigid geometry I. arXiv 1308.4734v5.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, vol. 93, Cambridge University Press.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory—Selected Topics in Point-Set Topology, New Mathematical Monographs, vol. 22, Cambridge University Press.Google Scholar
Henkin, L. (1950). A problem on inverse mapping systems. Proceedings of the American Mathematical Society 1 224–225.CrossRefGoogle Scholar
Jones, C. (1990). Probabilistic Non-Determinism. Phd thesis, University of Edinburgh. Technical Report ECS-LFCS-90-105.Google Scholar
Jones, C. and Plotkin, G. (1989). A probabilistic powerdomain of evaluations. In: Proceedings of the 4th Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, 186–195.CrossRefGoogle Scholar
Keimel, K. and Lawson, J. (2005). Measure extension theorems for T 0-spaces. Topology and its Applications 149 (1–3) 57–83.CrossRefGoogle Scholar
Keimel, K. and Lawson, J. (2009). d-completions and the d-topology. Annals of Pure and Applied Logic 159 292–306.CrossRefGoogle Scholar
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer Verlag, Berlin.CrossRefGoogle Scholar
Lawson, J. D. (1982). Valuations on continuous lattices. In: Hoffmann, R.-E. (ed.) Mathematische Arbeitspapiere, vol. 27, Universität Bremen, 204–225.Google Scholar
Prohorov, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theory of Probabilities and Applications 1 156–214.Google Scholar
Saheb-Djahromi, N. (1980). Cpo’s of measures for nondeterminism. Theoretical Computer Science 12 19–37.CrossRefGoogle Scholar
Steenrod, N. E. (1936). Universal homology groups. American Journal of Mathematics 58 (4) 661–701.CrossRefGoogle Scholar
Stone, A. H. (1979). Inverse limits of compact spaces. General Topology and its Applications 10 203–211.CrossRefGoogle Scholar
Tix, R. (1995). Stetige Bewertungen auf topologischen Räumen . Diplomarbeit, TH Darmstadt.Google Scholar
Waterhouse, W. C. (1972). An empty inverse limit. Proceedings of the American Mathematical Society 36 (2) 618.CrossRefGoogle Scholar