Lyapunov's Theorem for Measures on D-posets

Barbieri, Giuseppina

doi:10.1023/b:ijtp.0000048807.37145.cc

Cited by 20 publications

(12 citation statements)

References 13 publications

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“…Moreover, with the same proof as in 3.11 of [10] in the case n = 1, we can see that v(µ, ·) is a measure.…”

Section: The Spaces Pna and Dgsupporting

confidence: 56%

“…Recall that, if µ : L → R n is a nonatomic modular measure on a σ-complete D-lattice, by 3.12 of [10] µ(L) is convex. Hence we can give the following definition: let L be a σ-complete D-lattice, for a function v ∈ F 0 , we shall say that it is differentiable if it can be represented as v = f • µ for suitable f and µ such that µ : L → R n is a nonatomic σ-additive modular measure and f : µ(L) → R is a uniformly differentiable function which is null in zero.…”

Section: The Spaces Pna and Dgmentioning

confidence: 99%

“…Proof. Recall that, by 3.12 of [10] and (2.1)-6, µ(L) is bounded and convex. We can suppose that µ(L) has dimension m < n. Then we can find a linear map φ : R n → R m such that the restriction of φ to µ(L) is bijective.…”

Section: The Spaces Pna and Dgmentioning

confidence: 99%

“…Recall that, by 3.12 of [10], µ(L) is convex. Then we can consider the function φ defined as φ(t) = f ((jx + ty)/k) for t ∈ [0, 1].…”

Section: Theorem 34 There Exists An Aumann-shapley Operator φ On Pnmentioning

confidence: 99%

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

2006

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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.

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“…Moreover, with the same proof as in 3.11 of [10] in the case n = 1, we can see that v(µ, ·) is a measure.…”

Section: The Spaces Pna and Dgsupporting

confidence: 56%

Section: The Spaces Pna and Dgmentioning

confidence: 99%

Section: The Spaces Pna and Dgmentioning

confidence: 99%

“…Recall that, by 3.12 of [10], µ(L) is convex. Then we can consider the function φ defined as φ(t) = f ((jx + ty)/k) for t ∈ [0, 1].…”

Section: Theorem 34 There Exists An Aumann-shapley Operator φ On Pnmentioning

confidence: 99%

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

2006

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“…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. [4]). In Section 5, we have proved an analogue of the Lyapunov convexity theorem for a relatively non-atomic measure defined on a σ-complete effect algebra L.…”

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confidence: 99%

The range of non-atomic measures on effect algebras

Khare¹,

Singh²

2010

Demonstratio Mathematica

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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

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