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The sets of abstract probabilities. (English) Zbl 1133.60005

Adenier, Guillaume (ed.) et al., Foundations of probability and physics – 4, Växjö, Sweden, 4–9 June 2006. Proceedings of the Conference. Melville, NY: American Institute of Physics (AIP) (ISBN 978-0-7354-0391-8/hbk). AIP Conference Proceedings 889, 353-362 (2007).
It is an interesting attempt to generalize probability theory. In the author’s words: “One of the ways to extend our views on physical phenomena is to assign to observed events some nonclassical probabilities”. Earlier attempts using, e.g., complex numbers or \(p\)-adic numbers are mentioned. In the paper under review, the range of a probability is defined to be a set carrying a multiplication (commutative, associative, unit element, special idempotents) and a partial addition having natural properties (associativity, indecomposable zero element, distributivity, complementation, a version of the Archimedean property). Besides basic properties of such ranges (called probabilistic sets), some examples and probabilistic constructions are given.
For the entire collection see [Zbl 1115.81004].

MSC:

60A05 Axioms; other general questions in probability
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