Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-01T00:01:43.962Z Has data issue: false hasContentIssue false
  • English
  • Français

A normal form theorem for Lω1p, with applications

Published online by Cambridge University Press:  12 March 2014

Douglas N. Hoover
Douglas N. Hoover*
Affiliation:
Yale University, New Haven, Connecticut 06520
*
Queen's University, Kingston, Ontario K7L 3N6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 3 , September 1982 , pp. 605 - 624
Copyright
Copyright © Association for Symbolic Logic 1982

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

REFERENCES

[1]Aldous, D. J., Representations of partially exchangeable arrays of random variables, Journal of Multivariate Analysis, vol. 11 (1981).CrossRefGoogle Scholar
[2]Chung, K. L., A course in probability theory, 2nd edition, Academic Press, New York, 1974.Google Scholar
[3]Doob, J. L., Stochastic processes, Wiley, New York, 1953.Google Scholar
[4]Gaifman, H., Concerning measures in first order calculi, Israel Journal of Mathematics, vol. 2 (1964), pp. 1–18.CrossRefGoogle Scholar
[5]Hoover, D. N., Probability logic, Annals of Mathematical Logic, vol. 14 (1978), pp. 287–313.CrossRefGoogle Scholar
[6]Hoover, D. N., Model theory of probability logic, Ph.D. Thesis, University of Wisconsin-Madison, 1978.Google Scholar
[7]Keisler, H. J., Hyperfinite model theory, Logic Symposium '76 (Gandy, and Hyland, , Editors), North-Holland, Amsterdam, 1977.Google Scholar
[8]Kingman, J. F. C., Uses of exchangeability, Annals of Probability, vol. 6 (1978), pp. 183–197.CrossRefGoogle Scholar
[9]Krauss, P. H., Representations of symmetric probability models, this JournaL, vol. 34 (1969), pp. 183–193.Google Scholar
[10]Loeve, M., Probability theory, Van Nostrand, Princeton, New Jersey, 1963.Google Scholar
[11]Maharam, D., The representation of abstract integrals, Transactions of the American Mathematical Society, vol. 75 (1953), pp. 154–184.CrossRefGoogle Scholar
[12]Silverman, B. W., Limit theorems for dissociated random variables, Advances in Applied Probability, vol. 8 (1976), pp. 806–819.CrossRefGoogle Scholar