Finitistic and frequentistic approximation of probability measures with or without \(\sigma\)-additivity
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.
scientific article; zbMATH DE number 5319249
scientific article