Measure extension theorems for \(T_{0}\)-spaces. (English) Zbl 1152.06301
Summary: The theme of this paper is the extension of continuous valuations on the lattice of open sets of a \(T_0\)-space to Borel measures. A general extension principle is derived that provides a unified approach to a variety of extension theorems including valuations that are directed suprema of simple valuations, continuous valuations on locally compact sober spaces, and regular valuations on coherent sober spaces.
MSC:
| 06B35 | Continuous lattices and posets, applications |
| 28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |
References:
| [1] | Abramsky, S.; Jung, A., Domain theory, (Abramsky, S.; etal., Handbook of Logic in Computer Science, vol. 3 (1995), Clarendon Press: Clarendon Press Oxford) · Zbl 0876.68001 |
| [2] | M. Alvarez-Manilla, Measure theoretic results for continuous valuations on partially ordered spaces, Dissertation, Imperial College, London, 2000; M. Alvarez-Manilla, Measure theoretic results for continuous valuations on partially ordered spaces, Dissertation, Imperial College, London, 2000 |
| [3] | Alvarez-Manilla, M., Extension of valuations on locally compact sober spaces, Topology Appl., 124, 397-433 (2002) · Zbl 1062.28015 |
| [4] | Alvarez-Manilla, M.; Edalat, A.; Saheb-Djahromi, N., An extension result for continuous valuations, J. London Math. Soc. (2), 61, 629-640 (2000) · Zbl 0960.60005 |
| [5] | Berg, C.; Christensen, J. P.R.; Ressel, P., Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Graduate Texts in Mathematics, vol. 100 (1984), Springer: Springer Berlin, 289 pp · Zbl 0619.43001 |
| [6] | Billingsley, P., Probability and Measure (1986), Wiley: Wiley New York · Zbl 0649.60001 |
| [7] | Dellacherie, C.; Meyer, P.-A., Probabilities and Potential (1975), North-Holland: North-Holland Amsterdam |
| [8] | Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1088.06001 |
| [9] | C. Jones, Probabilistic non-determinism, Ph.D. thesis, University of Edinburgh, 1989; C. Jones, Probabilistic non-determinism, Ph.D. thesis, University of Edinburgh, 1989 |
| [10] | K. Keimel, R. Tix, Valuations and measures, Technische Universität Darmstadt, 1996, unpublished manuscript; K. Keimel, R. Tix, Valuations and measures, Technische Universität Darmstadt, 1996, unpublished manuscript |
| [11] | Lawson, J., Valuations on continuous lattices, (Hofmann, R.-E., Math. Arbeitspapiere, vol. 27 (1982), University of Bremen), 204-225 |
| [12] | Neveu, J., Bases Mathématiques du Calcul des Probabilités (1964), Masson & Cie: Masson & Cie Paris · Zbl 0137.11203 |
| [13] | Norberg, T., Existence theorems for measures on continuous posets, with applications to random set theory, Math. Scand., 64, 15-51 (1989) · Zbl 0667.60001 |
| [14] | Norberg, T.; Vervaat, W., Capacities on non-Hausdorff spaces, (Vervaat, W.; Holwerda, H., Probability and Lattices. Probability and Lattices, CWI Tract, vol. 110 (1997), CWI: CWI Amsterdam), 133-150 · Zbl 0883.28002 |
| [15] | Topsøe, F., Topology and Measure, Lecture Notes in Mathematics, vol. 133 (1970), Springer: Springer Berlin, xiv+79pp · Zbl 0197.33301 |
| [16] | Topsøe, F., Compactness in spaces of measures, Studia Math., 36, 195-212 (1970) · Zbl 0201.06202 |
| [17] | Topsøe, F., On construction of measures, (Proceedings of the Conference “Topology and Measures I”, Zinnowitz 1974 (1978), Greifswald University), 343-381 · Zbl 0412.28002 |
| [18] | G. Weidner, Bewertungen auf stetigen Verbänden, Master’s thesis, Technische Universität Darmstadt, 1995; G. Weidner, Bewertungen auf stetigen Verbänden, Master’s thesis, Technische Universität Darmstadt, 1995 |
| [19] | Wilker, P., Adjoint product and hom functors in general topology, Pacific J. Math., 34, 269-283 (1970) · Zbl 0205.52703 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
