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On measurable spaces and measurable maps. (English) Zbl 1112.06005

Summary: The author introduces and studies the category ID whose objects are suitable convergence D-posets of maps into the closed unit interval \(I\) and whose morphisms are sequentially continuous D-homomorphisms. It is shown that ID is dual to a subcategory of the category MID of generalized measurable spaces and generalized measurable maps. The author constructs epireflective and monocoreflective subcategories of ID and MID corresponding to two important properties of objects in ID, soberness and sequential closedness. The subcategories play important roles in applications to probability. Some basic probability notions are generalized so that the generalized random variables are dual to generalized observables and generalized probability measures are ID-morphisms. Let \(I_u\) be an ultrapower of \(I\). Some of the previous results are modified replacing \(I\) by \(I_u\) and replacing the sequential convergence by the approximation: a sequence approximates a point in \(I_n\) whenever the sequence of standard parts converges to the standard part of the point in \(I\).

MSC:

06A11 Algebraic aspects of posets
28E10 Fuzzy measure theory
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
60A05 Axioms; other general questions in probability
03H05 Nonstandard models in mathematics
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