Applications of Topological Dualities to Measure Theory in Algebraic Many-valued Logic

摘要

The problem of generalizing probability and measure theory to events described by non-classical, many-valued logics has recently attracted considerable attention from several research groups. A given non-classical logic yields via the Tarski-Lindenbaum construction a family of algebraic structures that generalize Boolean algebras. The elements of such an algebraic structure can then be thought of as generalized events. One seeks an appropriate notion of probability assignment-or state, following a terminology borrowed from operator algebras that has become standard for MV-algebras-to such generalized events. Axiomatic definitions of states attempt to generalize Kolmogorov's axioms, but often only require some form of finite additivity. In the presence of a sufficiently well-developed topological duality theory for the algebraic structures at hand, one can then attempt to represent such states as generalized integral operators acting on dual spaces. Representation results of this sort are modelled after the classical Riesz Representation Theorem. The generalized measures that dually correspond to states in this manner often satisfy some form of countable additivity, even if states themselves are only finitely additive. One thus sees that there is a fruitful interaction between topological dualities and probability theories of non-classical events. The 2008 edition of the biennial conference series ManyVal, titled Applications of Topological Dualities to Measure Theory in Algebraic Many-Valued Logic, explicitly focused on this interaction. This special issue, as a follow-up to that meeting, collects a selection of invited papers on the same topics. All contributions were refereed up to the journal's standards.