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.
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Schurz, G., Leitgeb, H. Finitistic and Frequentistic Approximation of Probability Measures with or without σ-Additivity. Stud Logica 89, 257–283 (2008). https://doi.org/10.1007/s11225-008-9128-3
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DOI: https://doi.org/10.1007/s11225-008-9128-3