Measures on MV-algebras

Abstract

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 SM(I). We describe its epireflective subcategory ASM(I) consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished σ-completion σH(A). The epireflection is the maximal object in 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 σH(A) depending on the choice of H. We show that the coproducts in the category of D-posets [9] 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 σH(A) generalizes the embedding of a field of events A into the generated σ-field σ(A), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8, 11, 14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the Loomis–Sikorski Theorem, objects in 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.