Convergence theorems for monotone measures

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

doi:10.1016/j.fss.2015.05.017

Cited by 45 publications

(16 citation statements)

References 80 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

See 1 more Smart Citation

Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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-…”

mentioning

confidence: 99%

Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

Self Cite

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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%