Convergence theorems for monotone measures

Li, Jun; Mesiar, Radko; Pap, Endre; Klement, Erich Peter

doi:10.1016/j.fss.2015.05.017

Cited by 43 publications

(16 citation statements)

References 80 publications

(185 reference statements)

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“…We point out that there are three important structural characteristics of monotone set function: strong order continuity [26] (see also [30]), the property (S) and the condition (E) [27].…”

Section: Discussionmentioning

confidence: 99%

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Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

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Given a pan-space () additivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations () , ⊕ ⊗ covers the Sugeno integral (based on () , ∨ ∧) and the Shilkret integral (based on () , ∨ ⋅), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.

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“…We point out that there are three important structural characteristics of monotone set function: strong order continuity [26] (see also [30]), the property (S) and the condition (E) [27].…”

Section: Discussionmentioning

confidence: 99%

“…It has been shown that Lebesgue's theorem, Riesz's theorem and Egoroff's theorem in classical measures theory ([20] [31]) hold in the case of monotone measures if and only if the monotone measures possess strong order continuity [26] [27][28] (see also[30]). We state these three con-…”

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confidence: 99%

Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

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“…This important theorem was generalized to monotone measure spaces. In [8], the concept of property (S) of monotone measures was introduced, and it was shown that the conclusion of the classical Riesz theorem holds for a monotone measure µ if and only if µ has property (S) ([8], also see [5,7,21]). We recall these results.…”

Section: Null-continuity and Convergence Of A Sequence Of Measurable mentioning

confidence: 99%

On Null-Continuity of Monotone Measures

2020

Mathematics

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The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures.

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“…Similarly, several convergence theorems known for the Lebesgue integral can be extended to the Choquet integral (for an overview, see [35] in this issue). For instance, for monotone measures satisfying some continuity property several versions of the Lebesgue monotone convergence theorem (Beppo Levi theorem) hold for the corresponding Choquet integral [47].…”

Section: The Choquet Integralmentioning

confidence: 99%

“…Later on, several not necessarily additive set functions were considered and studied in detail (see [54,74] for a bibliography, and also [35]). …”

Section: Introductionmentioning

confidence: 99%