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.

A novel approach to measuring uncertainty and uncertainty-based information in Dempster-Shafer theory is proposed (independently also proposed by Maeda et al. [1993]). It is shown that the proposed measure of total uncertainty in Dempster-Shafer theory is both additive and subadditive, has a desired range, and collapses correctly to either the Shannon entropy or the Hartley measure of uncertainty for special probability assignment functions. The paper is restricted, for the sake of simplicity, to finite sets.

Fuzzy Measure Theory

Measuring total uncertainty in dempster-shafer theory: a novel approach

Possibility theory was coined by L.A. Zadeh in the late seventies as an approach to model flexible restrictions constructed from vague pieces of information, described by means of fuzzy sets. Possibility theory is also a basic non-classical theory of uncertainty, different from but related to probability theory. This chapter discusses the basic elements of the theory: possibility and necessity measures (as well as two other set functions associated with a possibility distribution), the minimal specificity principle which underlies the whole theory, the notions of possibilitic conditioning and possibilistic independence, the combination, and projection of joint possibility distributions, as well as the possibilistic counterparts to integration. The relations and differences between this approach and other uncertainty frameworks, and especially probability theory, are pointed out. The difference between probability theory and fuzzy set theory is thus tentatively clarified. Lastly, decision-theoretic justifications of possibility theory are given.

Possibility Theory, Probability and Fuzzy Sets Misunderstandings, Bridges and Gaps

The Lebesgue integral is a monotone nonnegative linear function on the space of bounded measurable functions; its construction is strongly linked with the additivity of the underlying measure. However, the additivity of a measure is rather restrictive when modeling several real situations. To overcome this problem, several generalizations of classical probability measures have been proposed, such as pseudoadditive measures, possibility measures, belief functions, k -monotone capacities, and k -order additive measures. All these generalizations are covered by fuzzy measures. These set functions are sometimes called also premeasures. The Choquet integral is defined for nonnegative functions, and it is comonotone additive and homogeneous with respect to the multiplication by nonnegative reals. The Sugeno integral deals with functions whose range is contained with fuzzy sets and with normed fuzzy measures. The properties of the Choquet and the Sugeno integrals are recalled, discussed, and compared in this chapter. The chapter introduces a class of general fuzzy integrals with respect to a general fuzzy measure that has properties similar to those of Choquet or Sugeno integrals. To express the comonotone additivity of the Choquet integral and the comonotone maxitivity of the Sugeno integral, it is convenient to introduce a representation by means of a system of comonotone basic functions. The pseudoadditions and pseudomultiplications are also elaborated in the chapter.

CHAPTER 33 – Monotone Set Functions-Based Integrals

In this paper, I provide the basis for a measure- and integral-theoretic formulation of possibility theory. It is shown thai, using a general definition of possibility measures, and a generalization of Sugeno's fuzzy integral-the semi-normed fuzzy integral, or possibility integral-. a unified and consistent account can be given of many of the possibilistic results extant in the literature. The striking formal analogy between this treatment of possibility theory, using possibility integrals, and Kolmogorov's measure-theoretic formulation of probability theory, using Lebesgue integrals, is explored and exploited. I introduce and study possibilistic and fuzzy variables as possibilistic counterparts of stochastic and real stochastic variables respeclively, and develop the notion of a possibility distribution for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a Radon-Nikodym-like theorem in possibility theory. Following t...

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Fuzzy Measure Theory

Citations

Measuring total uncertainty in dempster-shafer theory: a novel approach

Possibility Theory, Probability and Fuzzy Sets Misunderstandings, Bridges and Gaps

CHAPTER 33 – Monotone Set Functions-Based Integrals

Possibility theory i: the measure- and integral-theoretic groundwork

Picture Fuzzy Hamacher Aggregation Operators and their Application to Multiple Attribute Decision Making

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