Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-01T01:16:41.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic convergence spaces

Part of: Foundations of probability theory Generalities Spaces with richer structures

Published online by Cambridge University Press:  09 April 2009

G. D. Richardson  and
D. C. Kent
G. D. Richardson
Affiliation:
Department of Mathematics University of Central FloridaOrlando, FL 32816, USA
D. C. Kent
Affiliation:
Department of Pure and Applied Mathematics Washington State UniversityPullman, WA 99164-3113, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A basic theory for probabilistic convergence spaces based on filter convergence is introduced. As in Florescu's previous theory of probabilistic convergence structures based on nets, one is able to assign a probability that a given filter converges to a given point. Various concepts and theorems pertaining to convergence spaces are extended to the realm of probabilistic convergence spaces, and illustrated by means of examples based on convergence in probability and convergence almost everywhere. Diagonal axioms due to Kowalsky and Fischer are also studied, first for convergence spaces and then in the setting of probabilistic convergence spaces.

Keywords

Probabilistic convergence spaceinitial probabilistic convergence structureprobabilistic closure operatorKowalsky diagonal axiomFischer diagonal axiom

MSC classification

Secondary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 54E70: Probabilistic metric spaces 60A05: Axioms; other general questions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 3 , December 1996 , pp. 400 - 420
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Chung, K. L., A course in probability theory (Academic Press, New York, 1974).Google Scholar
[2]Fischer, H. R., ‘Limesräume’, Math. Ann. 137 (1959), 269–303.CrossRefGoogle Scholar
[3]Florescu, L. C., ‘Probabilistic convergence structures’, Aequationes Math. 38 (1989), 123–145.CrossRefGoogle Scholar
[4]Frank, M. J., ‘Probabilistic topological spaces’, J. Math. Anal. and Appl. 34 (1971), 67–81.CrossRefGoogle Scholar
[5]Kent, D. C., ‘Convergence groups and convergence uniformities’, Fund. Math. 60 (1967), 213–222.CrossRefGoogle Scholar
[6]Kent, D. C., ‘Decisive convergence spaces, Fréchet spaces, and sequential spaces’, Rocky Mountain J. Math. 1 (1971), 367–374.CrossRefGoogle Scholar
[7]Kowalsky, H. J., ‘Limesräume und Komplettierung’, Math. Nachr. 12 (1954), 301–340.CrossRefGoogle Scholar
[8]Schweizer, B. and Sklar, A., Probabilistic metric spaces (North-Holland, New York, 1983).Google Scholar
[9]Sherwood, H., ‘On E-spaces and their relation to other classes of probabilistic spaces’, J. London Math. Soc. 44 (1969), 441–448.CrossRefGoogle Scholar