## 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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