On Weak Null-Additivity of Monotone Measures

“…The characteristics of strong absolute continuity of Type I and Type VI of fuzzy measures have been described by using convergence of sequence of measurable functions. As we have seen, such descriptions were done in a more general context concerning a pair of monotone measures, the previous related results [3,15] become to be some special cases of our new results.…”

Section: Discussionmentioning

confidence: 64%

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

2020

Mathematics

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In this note, the convergence of the sum of two convergent sequences of measurable functions is studied by means of two types of absolute continuity of fuzzy measures, i.e., strong absolute continuity of Type I, and Type VI. The discussions of convergence a.e. and convergence in measure are done in the general framework relating to a pair of monotone measures, and general results are shown. The previous related results are generalized.

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“…By using the fact that null-additivity implies property (S) on S-compact spaces (Proposition 8), we can obtain the following results (see also [25]). A) be an S-compact space and µ ∈ M be strongly order continuous.…”

Section: Definition 4 ([8])mentioning

confidence: 96%

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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On Weak Null-Additivity of Monotone Measures

Cited by 5 publications

References 20 publications

Convergence theorems for monotone measures

Convergence theorems for monotone measures

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

On Null-Continuity of Monotone Measures

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