scholarly journals Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mehmet Şahin ◽  
Necati Olgun ◽  
F. Talay Akyıldız ◽  
Ali Karakuş
Keyword(s):  
Measure Space ◽  
Extension Theorem ◽  
Fuzzy Measure ◽  
Outer Measure ◽  
Fuzzy Measures ◽  
Set Functions ◽  
Bottom Element ◽  
Entire Universe ◽  
Carathéodory Extension Theorem

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.

scholarly journals On the Analysis and Computation of Topological Fuzzy Measure in Distributed Monoid Spaces

Symmetry ◽  
2018 ◽  
Vol 11 (1) ◽  
pp. 9 ◽  
Author(s):  
Susmit Bagchi
Keyword(s):  
Fuzzy Sets ◽  
Measure Space ◽  
Fuzzy Measure ◽  
Topological Spaces ◽  
Multivalued Logic ◽  
Fuzzy Measures ◽  
Fuzzy Topological Spaces

The computational applications of fuzzy sets are pervasive in systems with inherent uncertainties and multivalued logic-based approximations. The existing fuzzy analytic measures are based on regularity variations and the construction of fuzzy topological spaces. This paper proposes an analysis of the general fuzzy measures in n-dimensional topological spaces with monoid embeddings. The embedded monoids are topologically distributed in the measure space. The analytic properties of compactness and homeomorphic, as well as isomorphic maps between spaces, are presented. The computational evaluations are carried out with n = 1, considering a set of translation functions with different symmetry profiles. The results illustrate the dynamics of finite fuzzy measure in a monoid topological subspace.

On the Fusion of Multiple Measure Based Belief Structures

2018 ◽  
Vol 26 (Suppl. 2) ◽  
pp. 63-88 ◽  
Author(s):  
Ronald R. Yager
Keyword(s):  
Fuzzy Measure ◽  
Uncertain Information ◽  
Fuzzy Measures ◽  
Belief Structures ◽  
Multiple Measure

We introduce the concept of a fuzzy measure and describe the process of combining fuzzy measures to form new measures. We discuss the role of fuzzy measures in modeling uncertain information and its use in modeling granular uncertain information with the aid of measure based belief structures. We turn to the problem of fusing multiple measure based belief structures. First we look at the case when the belief structures being fused have the same focal elements. Then we turn to case where the structures being fused have different focal elements. Finally we compare measure-based fusion with Dempster’s rule.

MEASURES OF LOCAL ENTROPIES FOR COMPOSITIVE FUZZY PARTITIONS

2011 ◽  
Vol 19 (04) ◽  
pp. 717-728 ◽  
Author(s):  
DORETTA VIVONA ◽  
MARIA DIVARI
Keyword(s):  
Functional Equations ◽  
Fuzzy Measure ◽  
System Of Functional Equations ◽  
Locality Property ◽  
Fuzzy Measures ◽  
Fuzzy Partitions

The aim of this paper is to characterize of the measures of entropies without probability or fuzzy measure for compositive fuzzy partitions, taking into account the so-called locality property. We propose a system of functional equations, whose solutions give some forms of entropies without probability or fuzzy measures.

Measures Defined by Gages

1992 ◽  
Vol 44 (6) ◽  
pp. 1303-1316 ◽  
Author(s):  
Washek F. Pfeffer ◽  
Brian S. Thomson
Keyword(s):  
Set Theory ◽  
Measure Space ◽  
Hausdorff Space ◽  
Outer Measure ◽  
Riesz Representation Theorem ◽  
Locally Compact ◽  
Riemann Integral ◽  
Riesz Representation ◽  
Intuitive Concept ◽  
Locally Compact Hausdorff Space

AbstractUsing ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

Fuzzy Measures: A solution to deal with community detection problems for networks with additional information

2020 ◽  
Vol 39 (5) ◽  
pp. 6217-6230
Author(s):  
Inmaculada Gutiérrez ◽  
Daniel Gómez ◽  
Javier Castro ◽  
Rosa Espínola
Keyword(s):  
Community Detection ◽  
Weighted Graph ◽  
Fuzzy Relation ◽  
Fuzzy Measure ◽  
Fuzzy Measures ◽  
Additional Information ◽  
Fuzzy Graphs ◽  
Finite Set ◽  
Definition Of ◽  
Community Detection Problems

In this work we introduce the notion of the weighted graph associated with a fuzzy measure. Having a finite set of elements between which there exists an affinity fuzzy relation, we propose the definition of a group based on that affinity fuzzy relation between the individuals. Then, we propose an algorithm based on the Louvain’s method to deal with community detection problems with additional information independent of the graph. We also provide a particular method to solve community detection problems over extended fuzzy graphs. Finally, we test the performance of our proposal by means of some detailed computational tests calculated in several benchmark models.

scholarly journals Statistical Parameters Based on Fuzzy Measures

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2015
Author(s):  
Fernando Reche ◽  
María Morales ◽  
Antonio Salmerón
Keyword(s):  
Product Space ◽  
Fuzzy Measure ◽  
Joint Analysis ◽  
Product Spaces ◽  
Statistical Parameters ◽  
Fuzzy Measures ◽  
Reference Set

In this paper, we study the problem of defining statistical parameters when the uncertainty is expressed using a fuzzy measure. We extend the concept of monotone expectation in order to define a monotone variance and monotone moments. We also study parameters that allow the joint analysis of two functions defined over the same reference set. Finally, we propose some parameters over product spaces, considering the case in which a function over the product space is available and also the case in which such function is obtained by combining those in the marginal spaces.

scholarly journals Construction of Fuzzy Measures over Product Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1605 ◽  
Author(s):  
Fernando Reche ◽  
María Morales ◽  
Antonio Salmerón
Keyword(s):  
Product Space ◽  
Fuzzy Measure ◽  
Product Measure ◽  
Product Spaces ◽  
Joint Function ◽  
Single Product ◽  
Fuzzy Measures ◽  
Statistical Indices ◽  
Product Measures ◽  
Definition Of

In this paper, we study the problem of constructing a fuzzy measure over a product space when fuzzy measures over the marginal spaces are available. We propose a definition of independence of fuzzy measures and introduce different ways of constructing product measures, analyzing their properties. We derive bounds for the measure on the product space and show that it is possible to construct a single product measure when the marginal measures are capacities of order 2. We also study the combination of real functions over the marginal spaces in order to produce a joint function over the product space, compatible with the concept of marginalization, paving the way for the definition of statistical indices based on fuzzy measures.

Borel-Cantelli Lemma for Sugeno Measure

2014 ◽  
Vol 614 ◽  
pp. 367-370 ◽  
Author(s):  
Chun Qin Zhang ◽  
Hui Zhang
Keyword(s):  
Measure Space ◽  
Natural Extension ◽  
Fuzzy Measure ◽  
Cantelli Lemma ◽  
Sugeno Measure

Sugeno measure is a fuzzy measure. In this paper, we derive the Borel-Cantelli lemma for Sugeno measure. This result is a natural extension of the classical Borel-Cantelli lemma to the case where the measure tool is fuzzy. The properties of Sugeno measure are further discussed. Then the Borel-Cantelli lemma will be proven on Sugeno measure space. This work generalizes the research and applications of the Borel-Cantelli lemma.

scholarly journals Some remarks concerning finitely Subadditive outer measures with applications

1998 ◽  
Vol 21 (4) ◽  
pp. 653-669 ◽  
Author(s):  
John E. Knight
Keyword(s):  
General Theory ◽  
Outer Measure ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Set Functions ◽  
Relationship Of ◽  
The Relationship ◽  
General Method

The present paper is intended as a first step toward the establishment of a general theory of finitely subadditive outer measures. First, a general method for constructing a finitely subadditive outer measure and an associated finitely additive measure on any space is presented. This is followed by a discussion of the theory of inner measures, their construction, and the relationship of their properties to those of an associated finitely subadditive outer measure. In particular, the interconnections between the measurable sets determined by both the outer measure and its associated inner measure are examined. Finally, several applications of the general theory are given, with special attention being paid to various lattice related set functions.

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