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A Unified Approach to Convergence Theorems of Nonlinear Integrals

  • Jun KawabeEmail author
Chapter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 22)

Abstract

There are several types of nonlinear integrals with respect to nonadditive measures, such as the Choquet, Šipoš, Sugeno, and Shilkret integrals. In order to put those integrals into practical use and aim for application to various fields, it is indispensable to establish convergence theorems of such nonlinear integrals. However, they have individually been discussed for each of the integrals up to the present. In this article, several important convergence theorems of nonlinear integrals, such as the monotone convergence theorem, the bounded convergence theorem, and the Vitali convergence theorem, are formulated in a unified way regardless of the types of integrals.

Keywords

Nonadditive measure Nonlinear integral Integral functional Convergence theorem Perturbation 

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 17K05293.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Division of Mathematics and Physics, Faculty of EngineeringShinshu UniversityNaganoJapan

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