Positive Operators à la Aumann-Shapley on Spaces of Functions on D-Lattices

Abstract

Let L be a σ-complete D-lattice and BV the AL-space of all realvalued, null in zero, functions on L of bounded variation. We prove the existence of a continuous Aumann-Shapley operator φ on the closed subspace of BV generated by powers of nonatomic σ-additive positive modular measures on L. The integral representation of φ on a class of functions that correspond to measure games is also exhibited.

On the Aumann–Shapley value

“…The two approaches are known to be equivalent, according to [12,Theorem 1.3.4], so that in particular lattice-ordered effect algebras and D-lattices are two ways of considering the same thing. We shall privilege the concepts of D-poset and of D-lattice, in order to be consistent with the previous paper [6].…”

“…Here and in the previous paper [6], a generalization of that kind is accomplished in case the original algebra of subsets C is replaced by a σ-complete lattice-ordered effect algebra L, thus weakening the assumptions of [2] and [10] in a remarkable way. In particular, in [6] it is proved the existence of a value φ on pNA, the subspace of BV spanned by powers of non-atomic σ-additive positive modular measures on L; furthermore, an integral representation of φ on a class of measure games is obtained.…”

“…In particular, in [6] it is proved the existence of a value φ on pNA, the subspace of BV spanned by powers of non-atomic σ-additive positive modular measures on L; furthermore, an integral representation of φ on a class of measure games is obtained.…”

“…The present paper carries on the investigation, namely we consider the generalization of the space BV and of its subspace bv NA to the framework of modular measures on a σ-complete lattice-ordered effect algebra L and, starting from the result of [6], we prove the existence of a (continuous) value on this subspace.…”

The range of non-atomic measures on effect algebras

“…In this section, we have given a Jordan type decomposition theorem for a locally bounded real-valued measure m defined on L, followed by various properties in the context of functions m + , m − and |m| (cf. [3]). In Section 4, using the notion of an atom of a real-valued measure m [24,25], we have showed that the range of a locally bounded real-valued σ-additive, non-atomic function m on a D-lattice L is an interval (−m − (1), m + (1)); characterizations of nonatomicity of m are established and used in obtaining this result (cf.…”