模糊测度理论中的f -fuzzy Lebesgue-Radon-Nikodym定理与微分

IF 2.7 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2025-09-09 DOI:10.1016/j.fss.2025.109579

Abbas Ghaffari , Reza Chaharpashlou

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引用次数: 0

摘要

基于Ghaffari等人（2022）[2]的基础性工作，引入了一个新的模糊测度理论框架，并通过基于三角范数的可分解时间戳模糊数值测度进行了积分，本文对 -模糊Lebesgue-Stieltjes测度和 -模糊微分进行了全面研究。我们在模糊环境中建立了经典Lebesgue-Radon-Nikodym定理的类比，将绝对连续性和测度分解概念推广到模糊测度空间，其中值是模糊数，操作是通过连续三角范数构造的。此外，我们给出了欧几里德空间上的 -模糊Lebesgue测度的Lebesgue微分定理。我们的研究结果为模糊积分和模糊微分建立了一个稳健的分析框架，在控制理论、风险建模和模糊微分方程的研究中具有潜在的应用前景。

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The ⁎-fuzzy Lebesgue-Radon-Nikodym theorem and differentiation in fuzzy measure theory

Based on the foundational work by Ghaffari et al. (2022) [2], which introduced a new framework for fuzzy measure theory and integration via triangular norm-based decomposable time-stamped fuzzy number-valued measures, this paper provides a comprehensive study of ⁎-fuzzy Lebesgue-Stieltjes measures and ⁎-fuzzy differentiation. We establish an analogue of the classical Lebesgue-Radon-Nikodym theorem in the fuzzy context, extending absolute continuity and measure decomposition concepts to fuzzy measure spaces where values are fuzzy numbers and operations are constructed through continuous triangular norms. Moreover, we present the Lebesgue differentiation theorem for ⁎-fuzzy Lebesgue measures on Euclidean spaces. Our results establish a robust analytical framework for fuzzy integration and differentiation, with potential applications in control theory, risk modeling, and the study of fuzzy differential equations.

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

期刊介绍： Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies. In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.