Abstract
The pan-integrals are based on a special type of commutative isotonic semiring \((\overline{R}_+, \oplus , \otimes )\) and the monotone measures \(\mu \) defined on a measurable space \((X,\mathcal {A})\). On the other hand, based on a pan-addition \(\oplus \) each monotone measure \(\mu \) generates a new monotone measure \(\mu _{\oplus }\) which is called the \(\oplus \)-optimal measure (to \(\mu \) and \(\oplus \)). Such monotone measure \(\mu _{\oplus }\) is greater than or equal to \(\mu \) and it is super-\(\oplus \)-additive (i.e., \(\mu _{\oplus }(A\cup B) \ge \mu _{\oplus }(A)\oplus \mu _{\oplus }(B)\) whenever \(A,B\in \mathcal {A}\), \(A\cap B=\emptyset \)). In this note, we shall present some new properties of the pan-integral. It is shown that the pan-integral with respect to \(\mu \) coincides with the pan-integral with respect to \(\mu _{\oplus }\) on a given pan-space \((X,\mathcal {A},\mu ,\overline{R}_+,\oplus ,\otimes )\). As a special case of this result, we show that the \(\oplus \)-optimal measure derived from \(\mu \) is totally balanced for the pan-integrals.
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Acknowledgements
This research was partially supported by the National Natural Science Foundation of China (Grant No. 11371332 and No. 11571106) and the NSF of Zhejiang Province (No. LY15A010013).
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Li, J., Ouyang, Y., Yu, M. (2017). Pan-Integrals Based on Optimal Measures. In: Torra, V., Narukawa, Y., Honda, A., Inoue, S. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2017. Lecture Notes in Computer Science(), vol 10571. Springer, Cham. https://doi.org/10.1007/978-3-319-67422-3_5
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