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Fifty Years of Fuzzy Logic and its Applications pp 125–151Cite as

Fuzzy Measures and Integrals: Recent Developments

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 326))

Abstract

This paper surveys the basic notions and most important results around fuzzy measures and integrals, as proposed independently by Choquet and Sugeno, as well as recent developments. The latter includes bases and transforms on set functions, fuzzy measures on set systems, the notion of horizontal additivity, basic Choquet calculus on the nonnegative real line introduced by Sugeno, the extension of the Choquet integral for nonmeasurable functions, and the notion of universal integral.

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Notes

  1. 1.

    A chain from \(\varnothing \) to \(X\) is a sequence \(\varnothing =A_0,A_1,\ldots , A_q=X\) of sets in \({\mathcal F}\) such that \(A_0\subset A_1\subset \cdots \subset A_q\). Its length is \(q\), and the chain is maximal if no other chain from \(\varnothing \) to \(X\) contains it.

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Author information

Authors and Affiliations

  1. Paris School of Economics, University of Paris I, 106-112, Bd. de L’Hôpital, 75013, Paris, France

    Michel Grabisch

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  1. Michel Grabisch
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Correspondence to Michel Grabisch .

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Editors and Affiliations

  1. Department of Computer Science, Texas State University, San Marcos, Texas, USA

    Dan E. Tamir

  2. School of Computing and Information Sciences, Florida International University, Miami, Florida, USA

    Naphtali D. Rishe

  3. School of Computing and Information Sciences, The University of South Florida, Tampa, Florida, USA, and Florida International University, Miami, Florida, USA

    Abraham Kandel

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Grabisch, M. (2015). Fuzzy Measures and Integrals: Recent Developments. In: Tamir, D., Rishe, N., Kandel, A. (eds) Fifty Years of Fuzzy Logic and its Applications. Studies in Fuzziness and Soft Computing, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-19683-1_8

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  • DOI: https://doi.org/10.1007/978-3-319-19683-1_8

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