Choquet integral and fuzzy measures on locally compact space.

*(English)*Zbl 0977.28012Summary: The concept of outer regular fuzzy measures is proposed, and it is shown that a functional of certain type on the cone of positive continuous functions with compact supports is represented as a Choquet integral with respect to an outer regular fuzzy measure. It is also shown that the Choquet integral of positive continuous functions with compact supports is represented as a Lebesgue integral with the same integrands. This representation is a generalization of certain previous results of others, which are useful for computation of the upper and lower expected value.

##### MSC:

28E10 | Fuzzy measure theory |

28C99 | Set functions and measures on spaces with additional structure |

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\textit{M. Sugeno} et al., Fuzzy Sets Syst. 99, No. 2, 205--211 (1998; Zbl 0977.28012)

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##### References:

[1] | Bourbaki, N., Topologie generale, (1971), Hermann Paris · Zbl 0249.54001 |

[2] | de Campos Ibañez, L.M.; Bolaños Carmona, M.J., Representation of fuzzy measures through probabilities, Fuzzy sets and systems, 31, 23-36, (1989) · Zbl 0664.28012 |

[3] | Choquet, G., Lectures on analysis, (1976), W.A. Benjamin Inc MA, 3rd printing |

[4] | Denneberg, D., Non additive measure and integral, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0826.28002 |

[5] | Dellacherie, C., Quelques commentaires sur LES prolongement de capacités, (), 77-81 |

[6] | Huber, P.J.; Strassen, U., Minimax tests and the Neyman-Pearson lemma for capacities, Ann. statist., 1, 251-263, (1973) · Zbl 0259.62008 |

[7] | Murofushi, T.; Sugeno, M., An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy sets and systems, 29, 201-227, (1989) · Zbl 0662.28015 |

[8] | Murofushi, T.; Sugeno, M., A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. math. anal. appl., 159, 532-549, (1991) · Zbl 0735.28015 |

[9] | Nguyen, H.T.; Walker, E.A., On decision making using belief functions, (), 311-336 |

[10] | Schmeidler, D., Integral representation without additivity, (), 253-261 · Zbl 0687.28008 |

[11] | Sugeno, M., Theory of fuzzy integrals and its applications, () · Zbl 0316.60005 |

[12] | Wassermann, L.A., Some applications of belief functions to statistical inference, (), 2nd ed. |

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