The range of non-atomic measures on effect algebras

Abstract

Abstract. The present paper deals with the study of superior variation m + , inferior variation m − and total variation |m| of an extended real-valued function m defined on an effect algebra L; having obtained a Jordan type decomposition theorem for a locally bounded real-valued measure m defined on L, we have observed that the range of a non-atomic function m defined on a D-lattice L is an interval (−m − (1), m + (1)). Finally, after introducing the notion of a relatively non-atomic measure on an effect algebra L, we have proved an analogue of Lyapunov convexity theorem for this measure. IntroductionLet H be a Hilbert space and let S(H) be a partially ordered group of all bounded self-adjoint operators on H. Put E(H) = {A ∈ S(H) : 0 ≤ A ≤ I}. If a quantum mechanical system F is represented in the usual way by a Hilbert space H, then the elements of E(H) correspond to effects for F [29,30]. Effects are of significance in representing unsharp measurements or observations on the system F [10], and effect valued measures play an important role in stochastic quantum mechanics [1,36]. As a consequence, there have been a number of recent efforts to establish appropriate axioms for logics, algebras, or posets based on effects [13,19]. In 1992, Kôpka defined D-posets of fuzzy sets in [18], which is closed under the formations of differences of fuzzy sets, while studying the axiomatical systems of fuzzy sets. A generalization of such structures to an abstract partially ordered set, where the basic operation is the difference, yields a very general and, at the same time, a very simple structure called a D-poset. A common generalization of orthomodular lattices and M V -algebras is termed as effect algebras introduced by Bennett and