On Probability Domains II

Abstract

We continue our studies of the foundation of probability theory using elementary category theory. We propose a classification scheme of probability domains in terms of cogenerators and their algebraic and topological properties and use the scheme to describe the transition from classical to fuzzy probability. We show that Łukasiewicz tribes form a category of natural probability domains in which σ -fields of sets are "minimal" and measurable [0, 1]-valued functions are "maximal" objects. The maximal objects form an epireflective subcategory in which both the classical and fuzzy probability can be modelled. This leads to a better understanding of the transition.

Real functions and the extension of generalized probability measures II

“…[19]). Second, we can extend the domain of functions and such extensions have stochastic applications, e.g., when studying the duality between generalized random variables and observables [5], [7], [8], [10], [14]- [16], [26].…”

“…They generalize Boolean algebras, MV-algebras and other probability domains, and provide a category in which observables and states become morphisms [2], [11]. Recall that a D-poset is a partially ordered set with the greatest element 1 X , the least element 0 X , and a partial binary operation called difference, such that a b is defined iff b ≤ a, and the following axioms are assumed: [16], [17], i.e., systems X ⊆ I X carrying the coordinatewise partial order, coordinatewise convergence of sequences, containing the top and bottom elements of I X , and closed with respect to the partial operation difference defined coordinatewise. We always assume that X is reduced, i.e., for x, y ∈ X, x = y, there exists u ∈ X such that u(x) = u(y).…”

On D-posets of fuzzy sets

“…D-posets of the class are called domains of probability and model generalized random events. Sequentially continuous D-homomorphisms into C model generalized probability measures ( [14]). …”

Real Functions and the Extension of Generalized Probability Measures

“…The category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms ( [28]) provides a natural background in which various classes of functions into [0,1] are objects and generalized probability measures, observables (dual maps to generalized random variables) are morphisms, and the extension of sequentially continuous maps is intrinsic (categorical), for example, both the extension of measures and the transition from measures to integrals can be viewed as an epireflection ([8], [9], [15]). …”

“…An interested reader can find more information about generalized probability in [3], [15], [16], [32] and in references therein.…”