A unified approach to the convergence theorems of nonlinear integrals containing decomposition integrals by the perturbative method

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05293
• 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): 2017-04-01 – 2023-03-31
• Keywords: 非加法的測度 / 非線形積分 / 摂動法 / 積分の収束定理 / 積分汎関数 / Choquet積分 / Sugeno積分 / Shilkret積分

Outline of Final Research Achievements

Various nonlinear convergence theorems are formulated for the Choquet, Sugeno, and Shilkret integrals to deal with a variety of modes of convergence of a sequence of measurable functions (for instance, pointwise convergence, almost everywhere convergence, convergence in measure, almost uniform convergence, and strong convergence in measure). Then, it is clarified what characteristics should be imposed on nonadditive measures for these nonlinear integral convergence theorems to hold. In addition to considering the individual integrals, the above convergence theorems are formulated for general nonlinear integral functionals defined on the product of the space of nonadditive measures and the space of measurable functions. Their validity is then investigated in a unified manner using our perturbation method.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

この研究では，個別の非線形積分に対して積分収束定理の定式化とその成立性を考察するだけでなく，一般の積分汎関数に対する定式化も行い，新たに開発した摂動法による解析手法を用いて，その成立性を考察している点が研究の特色であり，類例のない研究方法である．また，非線形積分の収束定理は，工学などの応用分野では，システムの積算過程の頑健性，一貫性，非カオス性を保証する大事な性質である．この研究により，非線形積分の収束定理の理論が格段に整備され，確固たる数学的基盤に基づいた応用が可能となる．