Finitistic and frequentistic approximation of probability measures with or without \(\sigma\)-additivity
type: Item
instance of: scholarly article
title: Finitistic and frequentistic approximation of probability measures with or without \(\sigma\)-additivity
zbMATH Open document ID: 1162.60002
published in: Studia Logica
publication date: 2008-09-02T00:00:00Z
review text: A theory of finitistic and frequentistic approximations (\(f\)-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 \(D\subset\varphi(N)\). The limiting probability measure over \(D\) can always be extended to a probability measure over \(\varphi(N)\), but this measure is not always \(\sigma\)-additive. It is concluded that probability measures can be regarded as idealizations of limiting frequencies if and only if \(\sigma\)-additivity is not assumed as a necessary axiom for probabilities. It is proved that \(\sigma\)-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full \(f\)-approximations. It is also shown that every non-\(\sigma\)-additive probability measure is \(f\)-approximable, though neither canonically nor fully \(f\)-approximable. Finally, the obtained results are translated to probability measures on open or closed formulas of first-order languages.
reviewed by: Vakhtang V. Kvaratskhelia
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 60A10
zbMATH DE Number: 5319249
zbMATH Keywords: probability
zbMATH Keywords: finite additivity
zbMATH Keywords: \(\sigma\)-additivity
zbMATH Keywords: frequency theory
zbMATH Keywords: finitistic approximation
author: Gerhard Schurz
author: Hannes Leitgeb
MaRDI profile type: Publication
full work available at URL: s11225-008-9128-3
OpenAlex ID: W2164696544
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DOI: 10.1007/S11225-008-9128-3
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description: scientific article; zbMATH DE number 5319249