Studia Logica

, Volume 89, Issue 2, pp 257–283

Finitistic and Frequentistic Approximation of Probability Measures with or without σ-Additivity

Authors

    • Department of PhilosophyUniversity of Duesseldorf
  • H. Leitgeb
    • Departments of Philosophy and MathematicsUniversity of Bristol
Article

DOI: 10.1007/s11225-008-9128-3

Cite this article as:
Schurz, G. & Leitgeb, H. Stud Logica (2008) 89: 257. doi:10.1007/s11225-008-9128-3

Abstract

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system \({{D\subseteq\wp(N)}}\). The limiting probability measure over D can always be extended to a probability measure over \({{\wp(N)}}\), but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages.

Keywords

Probabilityfinite additivityσ-additivityfrequency theoryfinitistic approximation

Copyright information

© Springer Science+Business Media B.V. 2008