Nonlinear integrals in nonadditive measure theory and their study based on a perturbative method

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400130
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: KAWABE Jun 信州大学, 学術研究院工学系, 教授 (50186136)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 非加法的測度 / 非線形積分 / 摂動法 / 積分収束定理 / 擬加法的性質 / Choquet積分 / Sugeno積分 / Shilkret積分

Outline of Final Research Achievements

We may view nonlinear integrals such as the Choquet, the Sugeno and the Shilkret integrals as nonlinear integral functionals defined on the product of the space of all nonadditive measures and the space of all measurable functions. From this point of view, we discussed the monotone convergence theorem and the bounded convergence theorem for such nonlinear integrals, together with the portmanteau theorem for nonadditvie measures, and verified that they can be formulated independently of the type of the integrals by using the notion of perturbation of integral functional. This perturbation manages the small change of the functional value arising as
a result of adding small amounts to a measure and an integrand.

Free Research Field

測度論