Measures: continuity, measurability, duality, extension

Abstract

ABSTRACT. We discuss some basic ideas and survey some fundamental constructions related to measure (a real-valued map the domain of which is a set of measurable objects carrying a suitable structure and the map partially preserves the structure): continuity, measurability, duality, extension. We show that in the category ID of difference posets of fuzzy sets and sequentially continuous difference-homomorphisms these constructions are intrinsic. Further, basic notions of the probability theory have natural generalizations within ID.

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].…”

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]). …”