Theory of capacities

Choquet, Gustave

doi:10.5802/aif.53

Cited by 3,667 publications

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“…Choquet [8] With the fine topology determined by the cone ^ of all superharmonic functions X --> [0, -h °°], the space X is known to be a completely regular Baire space (follows from Theorem 5.1), but neither locally compact nor first countable (follows from Theorem 4.2). Furthermore, X is connected and locally connected in the fine topology.…”

Section: G\(y)= F G(xy)d\(x))mentioning

confidence: 99%

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The quasi topology associated with a countably subadditive set function

Fuglede¹

1971

Annales De L’institut Fourier

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Section: G\(y)= F G(xy)d\(x))mentioning

confidence: 99%

“…If every compact set is capacitable, that is, if c is a capacity in the original sense of Choquet [8, § 15], then dually c(nK^)=infc(K^) (8) for any downward directed family of compact sets K^ (cf. (6) above).…”

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confidence: 99%

The quasi topology associated with a countably subadditive set function

Fuglede¹

1971

Annales De L’institut Fourier

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“…Aggregation with the Choquet integral: In a second step, the CI (Choquet, 1953;Murofushi and Sugeno, 1989;Grabisch, 1996) was used to aggregate the different measures into the corresponding criteria. In order to combine measures (individual utilities calculated with MACBETH) into criteria using the CI, the first step was the capacity identification.…”

Section: Datamentioning

confidence: 99%

Validation of a multi-criteria evaluation model for animal welfare

Martin¹,

Czycholl²,

Buxadé

et al. 2017

Animal

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The aim of this paper was to validate an alternative multi-criteria evaluation system to assess animal welfare on farms based on the Welfare Quality ® (WQ) project, using an example of welfare assessment of growing pigs. This alternative methodology aimed to be more transparent for stakeholders and more flexible than the methodology proposed by WQ. The WQ assessment protocol for growing pigs was implemented to collect data in different farms in Schleswig-Holstein, Germany. In total, 44 observations were carried out. The aggregation system proposed in the WQ protocol follows a three-step aggregation process. Measures are aggregated into criteria, criteria into principles and principles into an overall assessment. This study focussed on the first two steps of the aggregation. Multi-attribute utility theory (MAUT) was used to produce a value of welfare for each criterion and principle. The utility functions and the aggregation function were constructed in two separated steps. The MACBETH (Measuring Attractiveness by a Categorical-Based Evaluation Technique) method was used for utility function determination and the Choquet integral (CI) was used as an aggregation operator. The WQ decision-makers' preferences were fitted in order to construct the utility functions and to determine the CI parameters. The validation of the MAUT model was divided into two steps, first, the results of the model were compared with the results of the WQ project at criteria and principle level, and second, a sensitivity analysis of our model was carried out to demonstrate the relative importance of welfare measures in the different steps of the multi-criteria aggregation process. Using the MAUT, similar results were obtained to those obtained when applying the WQ protocol aggregation methods, both at criteria and principle level. Thus, this model could be implemented to produce an overall assessment of animal welfare in the context of the WQ protocol for growing pigs. Furthermore, this methodology could also be used as a framework in order to produce an overall assessment of welfare for other livestock species. Two main findings are obtained from the sensitivity analysis, first, a limited number of measures had a strong influence on improving or worsening the level of welfare at criteria level and second, the MAUT model was not very sensitive to an improvement in or a worsening of single welfare measures at principle level. The use of weighted sums and the conversion of disease measures into ordinal scores should be reconsidered.

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“…We define UM(S) where the first two classes are defined to consist of the continuous and of the upper semicontinuous functions X → [0, ∞[ with compact support. Therefore the set functions ϕ had sometimes to be restricted to the downward τ continuous ones, that is, to the capacities in the sense of [2].…”

Section: Introduction and Fundamentalsmentioning

confidence: 99%

“…In Section 48 of his famous Theory of capacities [2] Gustave Choquet introduced a certain class of functionals with the flavour of an integral, but invented for another purpose connected with capacities and not at all for the sake of measure and integration. Yet the concept showed basic qualities in that other respect too: It was in the initial spirit of Lebesgue [10] to construct the integral via decomposition into horizontal strips rather than vertical ones, which had fallen into oblivion in the course of the 20th century, and was simpler and much more comprehensive than the usual constructions.…”

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confidence: 99%

The (sub/super)additivity assertion of Choquet

König

2003

Studia Math.

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Abstract. The assertion in question comes from the short final section in Theory of capacities of Choquet (1953/54), in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this operation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and similarly for superadditivity. His treatment of this point was more like an outline, and his proof limited to a rather narrow special case. Thus the proper context and scope of the assertion has remained open. In this paper we present a counterexample which shows that the initial context has to be modified, and then in a new context we prove a comprehensive theorem which fulfils all the needs that have turned up so far.In Section 48 of his famous Theory of capacities [2] Gustave Choquet introduced a certain class of functionals with the flavour of an integral, but invented for another purpose connected with capacities and not at all for the sake of measure and integration. Yet the concept showed basic qualities in that other respect too: It was in the initial spirit of Lebesgue [10] to construct the integral via decomposition into horizontal strips rather than vertical ones, which had fallen into oblivion in the course of the 20th century, and was simpler and much more comprehensive than the usual constructions. Thus in subsequent decades the concept developed into a universal one in measure and integration, called the Choquet integral . One could even wonder why the Choquet integral has not become the foundation for all of integration theory.But the fact that this did not happen had an immediate reason: The basic hardship with the Choquet integral is that it is a priori obscure whether and when it is additive, which one subdivides into subadditive and superadditive. To this issue Choquet contributed in his final section 54 a spectacular, because much more abstract idea: On certain lattice cones all submodular and positive-homogeneous real-valued functionals must be subadditive, and the same for super in place of sub. It is this assertion which forms the theme of the present paper (in what follows the two cases will be united via an obvious sub/super shorthand notation). The treatment of Choquet was more like an outline, and his proof limited to a rather narrow special case. While the Choquet integral has been explored in subsequent decades, the abstract assertion has remained unsettled up to now.In recent years the present author needed an assertion of this kind to further develop his extended Daniell-Stone and Riesz representation theorems [7]. In 1998 he obtained an intermediate result which was sufficient for that purpose [8]. In this paper we present a counterexample which shows that the initial context for the abstract assertion has to be modified, and then in a new context we prove a comprehensive theorem which fulfils all the needs that have turned up so far. For the basic step in the so-called finite situation there will be two proofs. One of them is a...

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Theory of capacities

Cited by 3,667 publications

References 0 publications

The quasi topology associated with a countably subadditive set function

The quasi topology associated with a countably subadditive set function

Validation of a multi-criteria evaluation model for animal welfare

The (sub/super)additivity assertion of Choquet

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