The Multisubmeasures And The Choquet Integral Of Set-valued Functions With Respect To Multisubmeasures

The countable additivity of measures describes the measurement problem under the ideal conditions or null-error situations. Nevertheless in practical applications, the condition of countable additivity is too strict to hold sufficiently. Factually, peo-ple have found that the error couldn't be avoided. Some situations such as subjective judgements or non-repeated experiments are non-additive essentially. Choquet inte-gral theory based on non-additive measures has been investigated comprehensively and has been applied in many fields, such as information fusion, multiple criterias decision making, image processing, data mining, pattern recognition, and so on. However, in some applications, often involving the uncertainty of measures, this un-certainty description was expressed by multisubmeasures. Therefore, the Choquet integral theory involving set-valued function based on the multisubmeasures is an important part of fuzzy analysis. Based on the description of properties of nmulti-submeasures, studies the Choquet integral of set-valued function with respect to he multisubmeasures and a few kinds of Choquet integral operators.First of all, the properties of multisubmeasures are described, such as null-additive, pseudometric, (s)properties, exhaustive, and the relationship between these properties of multisubmeasures is given. What's more, studies the Egoroff's theo-rem, Lebesgue's theorem, Riesz's theorem by use of these properties.Second, on the basis of in-depth investigating the Choquet integral of real func-tion with respect to the non-additive measure, even the Choquet integral of set-valued function with respect to the non-additive measure (or of the real function with respect to the multisubmeasure), the Choquet integral of a set-valued function with respect to a multisubmeasure is defined and discussed by using the real-valued Choquet integral of the set-valued function with respect to the non-additive mea-sure, and some basic properties are characterized. It shows that a lot of characters could be well kept to its primitive such as the weakly null-additive, null-additive, converse null-additive, the Pseudometric property and the Darboux property, and so on.Finally, with ragard to the calculation model of the Choquet integral of interval value function with respect to interval-valued fuzzy measure, gives four types calcu-lation rules of Choquet integral on the discrete set. Therefore, by use of the COWA operator, we can transform the interval-valued fuzzy measur and the interval-valued integrand function into real numbers, and then calculate the corresponding Choquet integral,and give the numerical example.