共查询到1条相似文献,搜索用时 0 毫秒
1.
We study sequentially continuous measures on semisimple M
V-algebras. Let A be a semisimple M
V-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz M
V-algebra structure and the natural sequential convergence. Each separating set H of M
V-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple M
V-algebras carrying an initial sequential convergence induced by a separating set of M
V-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous M
V-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 σ
H
(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 σ
H
(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 M
V-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 σ
H
(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.
Supported by VEGA Grant 2/7193/01 相似文献