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On observables. (English) Zbl 0987.81009

Summary: We propose a simple set-theoretic model of a generalized probability space admitting intrinsic incompatible events and incompatible observables. It is a coproduct in the category \({\mathcal D}\) of \(D\)-posets and \(D\)-homomorphisms each factor of which is a classical Kolmogorovian probability space. Since classical events, random functions, and probability measures can be treated within \({\mathcal D}\) in a canonical way, the Kolmogorovian model becomes a special case. We show that \(\sigma\)-additivity and other \(\sigma\)-notions can be replaced in a natural way by sequential continuity.

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
60A10 Probabilistic measure theory
60A99 Foundations of probability theory
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