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A Domain-Theoretic Approach to Integration in Hausdorff Spaces

Published online by Cambridge University Press:  01 February 2010

J. D. Howroyd
J. D. Howroyd
Affiliation:
Department of Mathematics, Goldsmiths College, University of London, New Cross, London SE14 6NW, [email protected], http://homepages.gold.ac.uk/jhowroyd/

Abstract

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In this paper we generalize the construction of a domain-theoretic integral, introduced by Professor Abbas Edalat, in locally compact separable Hausdorff spaces, to general Hausdorff spaces embedded in a domain. Our main example of such spaces comprises general metric spaces embedded in the rounded ideal completion of the partially ordered set of formal balls. We go on to discuss analytic subsets of a general Hausdorff space, and give a sufficient condition for a measure supported on an analytic set to be approximated by a sequence of simple valuations. In particular, this condition is always satisfied in a metric space embedded in the rounded ideal completion of its formal ball space. We finish with a comments section, where we highlight some potential areas for future research and discuss some questions of computability.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 229 - 273
Copyright
Copyright © London Mathematical Society 2000

References

1. Abramsky, S. and Jung, A., ‘Domain theory’, Handbook of logic in computer science, Vol. III (ed. Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E., Oxford University Press, 1994).Google Scholar
2. Edalat, A., ‘Domain of computation of a random field in statistical physics’, Theory and Formal Methods 1994: Proceedings of the Second Imperial College Workshop (Imperial College Press, 1995).Google Scholar
3. Edalat, A., ‘Domain theory and integration’, Theoret. Comput. Sci. 151 (1995) 163–193.CrossRefGoogle Scholar
4. Edalat, A., ‘Domain theory in stochastic processes’, Proc. Tenth Annual IEEE Symposium on Logic in Computer Science (LICS) (IEEE, 1995).CrossRefGoogle Scholar
5. Edalat, A., ‘Dynamical systems, measures and fractals via domain theory’, Inform and Comput., 120 (1995) 32–48.CrossRefGoogle Scholar
6. Edalat, A., ‘Power domains and iterated function systems’, Inform and Comput. 124 (1996) 182–197.CrossRefGoogle Scholar
7. Edalat, A., ‘Domains for computation in mathematics, physics and exact real arithmetic’, Bull. Symbolic Logic 3 (1997) 401–452.CrossRefGoogle Scholar
8. Edalat, A., ‘When Scott is weak on the top’, Math. Structures Comput. Sci. 7 (1997) 403’417.CrossRefGoogle Scholar
9. Edalat, A. and Escardó, M. H., ‘Integration in real PCF’, Proc. Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS) (IEEE, 1996).CrossRefGoogle Scholar
10. Edalat, A. and Heckmann, R., ‘A computational model for metric spaces’, Theoret. Comput. Sci. 193 (1998) 53–73.CrossRefGoogle Scholar
11. Edalat, A. and Krznarić, M., ‘Numerical integration with exact real arithmetic, Automata languages and programming, Lecture Notes in Comput. Sci. 1644 (Springer, 1999)90–104.CrossRefGoogle Scholar
12. Edalat, A. and Negri, S., ‘The generalised Riemann integral on locally compact spaces’, Topology Appl. 89 (1998) 121–150.CrossRefGoogle Scholar
13. Edalat, A. and Parry, J., An algorithm to estimate the Hausdorff dimension of selfaffine sets’, Electronic Notes in Theoret. Comput. Sci. 13 (1998).CrossRefGoogle Scholar
14. Edalat, A. and Sünderhauf, P., ‘A domain-theoretic approach to computability on the real line’, Theoret. Comput. Sci. 210 (1997) 73–98.CrossRefGoogle Scholar
15. Edalat, A. and Sünderhauf, P.. ‘Computable Banach spaces via domain theory’, The-oret. Comput. Sci. 219 (1999) 169–184.Google Scholar
16. Federer, H., Geometric measure theory, Grundlehren Math. Wiss. 153 (Springer-Verlag, 1969).Google Scholar
17. Heckmann, R., ‘Spaces of valuations’, Proc. Summer Conference on General Topology and Applications 1995, Ann. New York Acad. Sci. Vol. 806 (1996) 174–200.CrossRefGoogle Scholar
18. Heckmann, R., ‘Approximation of metric spaces by partial metric spaces’, Applied Categ. Structures 7 (1999) 71–83.CrossRefGoogle Scholar
19. Jones, C., ‘Probabilistic non-determinism, PhD thesis, University of Edinburgh, 1989.Google Scholar
20. Kechris, A. S., Classical descriptive set theory, Grad. Texts in Math. 156 (Springer-Verlag, 1995).Google Scholar
21. Kelley, J. L., General topology (Van Nostrand, 1955).Google Scholar
22. Kirch, O., ‘Bereiche und Bewertungen’, Master's thesis, Technische Hochschule Darmstadt, 1993.Google Scholar
23. König, H., Measure and integration (Springer-Verlag, 1997).Google Scholar
24. Lawson, J. D., ‘Spaces of maximal points’, Math. Structures Comput. Sci. 1 (1997) 543–555.CrossRefGoogle Scholar
25. Lawson, J.D., ‘Computation on metric spaces via domain theory’, Topology Appl. 85 (1998) 247–263.CrossRefGoogle Scholar
26. Pfeffer, W. F., Integrals and measures, Pure Appl. Math. 42 (Dekker, 1977).Google Scholar
27. Rogers, C. A., Hausdorff measures (Cambridge University Press, 1970).Google Scholar