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Construction of Fuzzy Measures over Product Spaces

Author

Listed:
  • Fernando Reche

    (Department of Mathematics and Center for the Development and Transfer of Mathematical Research to Industry (CDTIME), University of Almería, 04120 Almería, Spain)

  • María Morales

    (Department of Mathematics and Center for the Development and Transfer of Mathematical Research to Industry (CDTIME), University of Almería, 04120 Almería, Spain)

  • Antonio Salmerón

    (Department of Mathematics and Center for the Development and Transfer of Mathematical Research to Industry (CDTIME), University of Almería, 04120 Almería, Spain)

Abstract

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.

Suggested Citation

  • Fernando Reche & María Morales & Antonio Salmerón, 2020. "Construction of Fuzzy Measures over Product Spaces," Mathematics, MDPI, vol. 8(9), pages 1-18, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1605-:d:415050
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    References listed on IDEAS

    as
    1. Jun Li, 2020. "On Null-Continuity of Monotone Measures," Mathematics, MDPI, vol. 8(2), pages 1-13, February.
    2. Ghirardato, Paolo, 1997. "On Independence for Non-Additive Measures, with a Fubini Theorem," Journal of Economic Theory, Elsevier, vol. 73(2), pages 261-291, April.
    3. Hendon, Ebbe & Jacobsen, Hans Jorgen & Sloth, Birgitte & Tranaes, Torben, 1996. "The product of capacities and belief functions," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 95-108, October.
    4. Denneberg, Dieter, 2002. "Conditional expectation for monotone measures, the discrete case," Journal of Mathematical Economics, Elsevier, vol. 37(2), pages 105-121, April.
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    Cited by:

    1. Gia Sirbiladze & Tariel Khvedelidze, 2023. "Associated Statistical Parameters’ Aggregations in Interactive MADM," Mathematics, MDPI, vol. 11(4), pages 1-17, February.

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