Measures on circle coarse-grained systems of sets

Simone, Anna De; Pták, Pavel

doi:10.1007/s11117-009-0015-6

Cited by 7 publications

(9 citation statements)

References 11 publications

(8 reference statements)

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“…In extending states on a certain kind of quantum logics (see [2][3][4]; for some other related consideration, see also [8,10]), the results obtained are based on the (state) Horn-Tarski measure extension theorem (see [1], [3] and [9]). Let us call this theorem the HT proper.…”

Section: Introduction and Resultsmentioning

confidence: 99%

On the Farkas lemma and the Horn–Tarski measure-extension theorem

Simone

Pták

2015

Linear Algebra and its Applications

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MSC: 15A06 06E99 28A60 Keywords: Non-negative solutions of systems of linear equations The Farkas lemma Measure extension State on a Boolean algebra The Horn-Tarski criterionWe first derive a certain version of the Farkas lemma called the 0-1 Farkas lemma (the 0-1 FL). We then show that the 0-1 FL is equivalent to a measure-extension theorem. By applying one implication of this result, we prove that the 0-1 FL implies the classical Horn-Tarski measure-extension theorem.

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Section: Introduction and Resultsmentioning

confidence: 99%

On the Farkas lemma and the Horn–Tarski measure-extension theorem

Simone

Pták

2015

Linear Algebra and its Applications

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“…Extendability of probabilities on (pre-)Dynkin-systems has already been part of discussions in quantum probability since 1969 [Gudder, 1969] up to more current times [De Simone and Pták, 2010]. Several necessary and/or sufficient conditions on the structure of D and/or the values of µ are known [Gudder, 1984, De Simone et al, 2007, De Simone and Pták, 2010.…”

Section: Extendabilitymentioning

confidence: 99%

“…Extendability of Dynkin probability spaces (as well as for finitely additive probabilities on pre-Dynkin-systems) has already been part of discussions in quantum probability since 1969 [Gudder, 1969] up to more current times [De Simone and Pták, 2010]. Most of the results, e.g.…”

Section: A Lemmas and Proofsmentioning

confidence: 99%

“…Most of the results, e.g. [Gudder, 1984, De Simone et al, 2007, De Simone and Pták, 2010, are only stated in terms of finitely additive probability measure and/or apply only on a structurally restricted class of Dynkin-systems. We found a theorem in the literature on marginal probabilities which can be tweaked to give a universal sufficient and necessary criterion for σ-extendability.…”

Section: A Lemmas and Proofsmentioning

confidence: 99%

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The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

Derr¹,

Williamson²

2023

Preprint

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In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkinsystems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley [Walley, 1991, p. 84]. Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities. When posing problems in probability calculus,it should be required to indicate for which events the probabilities are assumed to exist.

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“…It should be noted that a collection of subsets of a set that is subject to the above requirements (i)–(iii) is sometimes called a (concrete) quantum logic (, , , , etc.). Our system

(Ω, D)

can be therefore seen, when interpreted within the algebraic foundation of quantum mechanics, as an enriched quantum logic.…”

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confidence: 99%

States on systems of sets that are closed under symmetric difference

2015

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We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allowfor extensions. Finally,we consider these questions in a σ-complete setup and find a large class of such systems with rather interesting state properties

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Measures on circle coarse-grained systems of sets

Abstract: We show that a (non-negative) measure on a circle coarse-grained system of sets can be extended, as a (non-negative) measure, over the collection of all subsets of the circle. This result contributes to quantum logic probability (de Lucia in

Cited by 7 publications

References 11 publications

On the Farkas lemma and the Horn–Tarski measure-extension theorem

On the Farkas lemma and the Horn–Tarski measure-extension theorem

The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

States on systems of sets that are closed under symmetric difference

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