Introduction. Required Background in Set Theory. Fuzzy Measures. Extensions. Structural Characteristics for Set Functions. Measurable Functions on Fuzzy Measure Spaces. Fuzzy Integrals. PanIntegrals. Applications. Index.

This chapter surveys the fundamental aspect of non-additive measures and integral with respect to a non additive measure. Several basic definitions of non additive measure.

Fuzzy Measure Theory

Integral with Respect to a Non Additive Measure: An Overview

A granular fuzzy set theory is modeled on fuzzy sets whose membership functions are defined on sets of sets (granules). The grade is interpreted literally; for example, that the grade of x is 1/2 means one half of the granule x belongs to the fuzzy set. By taking the union of these subgranules, one get a crisp set representation of a fuzzy set. In other words, a granular fuzzy set is a fuzzy set that has a crisp set representation. A measure theory based on such granular fuzzy sets is developed. The measure theory of crisp sets is imported to fuzzy sets via their crisp representations. Using such a notion, a belief function can be shown to be the inner probability of a probability theory of fuzzy sets.

http://ieeexplore.ieee.org/iel5/6323/16906/00781806.pdf

Measure theory on granular fuzzy sets

Parameter estimation for Choquet fuzzy integral based on Takagi–Sugeno fuzzy model

We study the relation between possibility measures and the theory of imprecise probabilities. It is shown that a possibility measure is a coherent upper probability iff it is normal. We also prove that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all positive gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that a supremum preserving upper prevision is not necessarily coherent! This makes us look for alternative extensions of possibility measures that are not necessarily supremum preserving, through natural extension.

/pdf/on-the-coherence-of-supremum-preserving-upper-previsions-49i490rq2i.pdf

On the Coherence of Supremum Preserving Upper Previsions

In this paper, a method for fusing fuzzy data relevant both to drivers' experience and provided information is presented. Expected travel time is then updated according to the results of fusion. The method takes into account the "compatibility" of data originating from different sources, and provides information about acceptability of results. Influence of uncertainty on drivers' compliance with provided information is examined in detail, according to uncertainty-based Information Theory.

Fuzzy Measure Theory

Citations

Integral with Respect to a Non Additive Measure: An Overview

Measure theory on granular fuzzy sets

Parameter estimation for Choquet fuzzy integral based on Takagi–Sugeno fuzzy model

On the Coherence of Supremum Preserving Upper Previsions

Fuzzy data fusion for updating information in modeling drivers' choice behavior

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