We present a categorical approach to the extension of probabilities, i.e. normed σ-additive measures. J. Novák showed that each bounded σ-additive measure on a ring of sets A is sequentially continuous and pointed out the topological aspects of the extension of such measures on A over the generated σ-ring σ(A): it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space X over its Čech-Stone compactification βX (or as the extension of continuous functions on X over its Hewitt realcompactification υX). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that σ(A) is the sequential envelope of A with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category ID of D-posets of fuzzy sets (such D-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on A over σ(A) is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems.
The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category ID of D-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within ID) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a "simplex'' modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon.
The ring $B(R)$ of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring $C(R)$ of all continuous functions and, similarly, the ring $\mathbb{B}$ of all Borel measurable subsets of $R$ is a sequential ring completion of the subring $\mathbb{B}_0$ of all finite unions of half-open intervals; the two completions are not categorical. We study $\mathcal L_0^*$-rings of maps and develop a completion theory covering the two examples. In particular, the $\sigma $-fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets $\mathbb{A}$, the generated $\sigma $-field $\sigma (\mathbb{A})$ yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative $\mathcal L_0^*$-groups.