Integrals of set-valued functions

Aumann, Robert J.

doi:10.1016/0022-247x(65)90049-1

Cited by 1,132 publications

(509 citation statements)

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“…Definition 2.1 ( [2,4,5]). Let X be a nonempty set, Ω be a σ-algebra of subsets of X, and µ : Ω → [0, ∞) be a nonnegative real-valued set function.…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

On Jensen-Type and Hölder-Type Inequality for Interval-Valued Choquet Integrals

Jang¹,

Lee²,

Kim³

2018

IJFIS

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“…Definition 2.1 ( [2,4,5]). Let X be a nonempty set, Ω be a σ-algebra of subsets of X, and µ : Ω → [0, ∞) be a nonnegative real-valued set function.…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…The Choquet integrals have been studied by many researchers (see [1][2][3][4][5][6]). Aumann [7], Jang and his colleagues [8][9][10][11][12][13], and Zhang et al [14] also have been studying the interval-valued Choquet integrals which are related with some properties and applications of them.…”

Section: Introductionmentioning

confidence: 99%

On Jensen-Type and Hölder-Type Inequality for Interval-Valued Choquet Integrals

Jang¹,

Lee²,

Kim³

2018

IJFIS

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“…As suggested by Aumann (1965), one can think of a random closed set (or random correspondence in Aumann's work) as a bundle of random variables -its measurable selections, see De…nition A.3…”

Section: Representation Through Random Sets Theorymentioning

confidence: 99%

“…The de…nition of the sharp identi…cation region in Assumption 2.5 indicates that one needs to take conditional expectations of the elements of Sel (Q ) : Observe that by Assumption 2.3, Q is an integrably bounded random closed set, and therefore all its selections are integrable. Hence, we can de…ne the conditional Aumann expectation (Aumann (1965)) of Q as…”

Section: Representation Through Random Sets Theorymentioning

confidence: 99%

Sharp identification regions in models with convex moment predictions

Beresteanu¹,

Molchanov²,

Molinari³

2010

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We provide a tractable characterization of the sharp identi…cation region of the parameters in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move …nite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for ; we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly de…ned random set. The sharp identi…cation region of ; denoted I ; can then be obtained as the set of minimizers of the distance from a properly speci…ed vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to e¢ ciently verify whether a candidate is in I : We use examples analyzed in the literature to illustrate the gains in identi…cation and computational tractability a¤orded by our method.

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“…Nine years later, Robert Aumann in [2] introduced a different definition of the multivalued integral. This concept was based on the Lebesgue integral for real functions.…”

Section: Introductionmentioning

confidence: 99%