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Extension of domains of states. (English) Zbl 1166.28006

Kolmogorov’s classical probability theory has been generalized in many ways. States, as understood in this paper, are generalized probability measures, whose domain is a set of generalized probability events carrying a certain algebraic structure (such as orthoalgebras, quantum logics, MV-algebras, effect algebras, D-posets, in particular, fields of sets in the classical setting) and with range the unit interval. For this paper general systems of fuzzy sets, i.e. systems of \([0,1]\)-valued maps, are peculiar, in particular bold algebras, Łukasiewicz tribes, and special classes of D-posets and effect algebras lie in the center of interest. The author proposes to extend his work on the extension of states to a larger domain by describing some general topological and categorical aspects of such extensions, being motivated by the pioneering work of Josef Novak on sequential envelopes. He believes that the category ID of D-posets of fuzzy sets is a suitable base for a categorical approach to the so-called fuzzy probability as developed by S. Bugajski [Int. J. Theor. Phys. 35, No. 11, 2229-2244 (1996; Zbl 0872.60003)] and S. Gudder [Demonstr. Math. 31, 235–254 (1998; Zbl 0952.60002)]. Since fields of sets form a distinguished subcategory of ID and probability measures are morphisms, it covers the classical case. This category, suitable subcategories and related categories are used to describe the relationship between \(\sigma\)-fields of crisp sets and generated Łukasiewicz tribes of measurable functions. The duality of D-posets of fuzzy sets and the corresponding measurable spaces yields the duality between observables and fuzzy random variables. It is proved, that some fuzzy variables can’t be modelled and interpreted within the classical probability theory. This means that the passage from crisp events to fuzzy events yields a genuine generalization.

MSC:

28E10 Fuzzy measure theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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