## Measures on MV-algebras.(English)Zbl 1021.28012

Summary: We study sequentially continuous measures on semisimple MV-algebras. Let $$A$$ be a semisimple MV-algebra and let $$I$$ be the interval $$[0,1]$$ carrying the usual Łukasiewicz MV-algebra structure and the natural sequential convergence. Each separating set $$H$$ of MV-algebra homomorphisms of $$A$$ into $$I$$ induces on $$A$$ an initial sequential convergence. Semisimple MV-algebras carrying an initial sequential convergence induced by a separating set of MV-algebra homomorphisms into $$I$$ are called $$I$$-sequential and, together with sequentially continuous MV-algebra homomorphisms, they form a category $$\mathcal{SM}(I)$$. We describe its epireflective subcategory $$\mathcal{ASM}(I)$$ consisting of absolutely sequentially closed objects and we prove that the epireflection sends $$A$$ into its distinguished $$\sigma$$-completion $$\sigma_H(A)$$. The epireflection is the maximal object in $$\mathcal{SM}(I)$$ which contains $$A$$ as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of $$\sigma_H(A)$$ depending on the choice of $$H$$. We show that the coproducts in the category of $$D$$-posets of suitable families of $$I$$-sequential MV-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: $$H$$ plays the role of elementary events, the embedding of $$A$$ into $$\sigma_H(A)$$ generalizes the embedding of a field of events $$A$$ into the generated $$\sigma$$-field $$\sigma(A)$$, and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras and, unlike in the Loomis-Sikorski Theorem, objects in $$\mathcal{ASM}(I)$$ correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, $$D$$-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature.

### MSC:

 28E10 Fuzzy measure theory 06D35 MV-algebras
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