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

Research Project

Project/Area Number 17K05293
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionShinshu 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
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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 / 摂動性 / 積分汎関数の収束定理 / 分布型積分 / 分割型積分 / Lehrer積分 / 包除積分 / 収束定理 / ショケ積分 / 実関数論 / 測度論 / 積分収束定理
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.

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.

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (35 results)

All 2020 2019 2018 2017 Other

All Journal Article (12 results) (of which Peer Reviewed: 7 results,  Open Access: 12 results) Presentation (20 results) (of which Int'l Joint Research: 7 results,  Invited: 6 results) Remarks (3 results)

  • [Journal Article] Convergence in measure theorems of nonlinear integrals of functions integrable to the pth power2020

    • Author(s)
      Jun Kawabe
    • Journal Title

      Fuzzy Sets and Systems

      Volume: - Pages: 29-41

    • DOI

      10.1016/j.fss.2019.12.007

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Convergence in measure theorems of the Choquet integral revisited2019

    • Author(s)
      Jun Kawabe
    • Journal Title

      Lecture Notes in Artificial Intelligence

      Volume: - Pages: 17-28

    • DOI

      10.1007/978-3-030-26773-5_2

    • ISBN
      9783030267728, 9783030267735
    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Choquet積分の測度収束定理の精密化2019

    • Author(s)
      河邊 淳
    • Journal Title

      第24回曖昧な気持ちに挑むワークショップ講演論文集

      Volume: - Pages: 28-33

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] 測度収束関数列の非線形積分のp次収束性2019

    • Author(s)
      河邊 淳
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2143 Pages: 51-61

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] The Vitali convergence in measure theorem of nonlinear integrals2019

    • Author(s)
      Jun Kawabe
    • Journal Title

      Fuzzy Sets and Systems

      Volume: - Pages: 63-81

    • DOI

      10.1016/j.fss.2019.04.003

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A unified approach to convergence theorems of nonlinear integrals2018

    • Author(s)
      Jun Kawabe
    • Journal Title

      Advances in Mathematical Economics

      Volume: 22 Pages: 93-116

    • DOI

      10.1007/978-981-13-0605-1_4

    • ISBN
      9789811306044, 9789811306051
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] 積分汎関数に対するVitali型収束定理2018

    • Author(s)
      河邊 淳
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 2095 Pages: 124-130

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] Convergence theorems of the Choquet integral for three types of convergence of measurable functions2018

    • Author(s)
      Jun Kawabe
    • Journal Title

      Josai Mathematical Monographs

      Volume: 11 Pages: 55-74

    • NAID

      120006487234

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The Vitali type theorem for the Choquet integral2017

    • Author(s)
      Jun Kawabe
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 3 Pages: 349-365

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] 非線形積分の収束定理の統一的定式化2017

    • Author(s)
      河邊 淳
    • Journal Title

      第56回実函数論・函数解析学合同シンポジウム講演集

      Volume: 56 Pages: 35-54

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] 対称・反対称Choquet積分に対するVitaliの収束定理2017

    • Author(s)
      河邊 淳
    • Journal Title

      実解析学シンポジウム2017講演集

      Volume: 49 Pages: 31-36

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] 一様Choquet可積分性とVitali型収束定理2017

    • Author(s)
      河邊 淳
    • Journal Title

      第22回曖昧な気持ちに挑むワークショップ講演論文集

      Volume: 22 Pages: 62-67

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] The Vitali convergence theorem of nonlinear integral functionals2019

    • Author(s)
      Jun Kawabe
    • Organizer
      The First Analysis Mathematica International Conference
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Convergence in measure theorems of the Choquet integral revisited2019

    • Author(s)
      Jun Kawabe
    • Organizer
      The 16th International Conference on Modeling Decisions for Artificial Intelligence
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] p次可積分関数列の非線形積分の収束定理2019

    • Author(s)
      河邊 淳
    • Organizer
      日本数学会2019年秋季総合分科会
    • Related Report
      2019 Research-status Report
  • [Presentation] Choquet積分の測度収束定理の精密化2019

    • Author(s)
      河邊 淳
    • Organizer
      第24回曖昧な気持ちに挑むワークショップ
    • Related Report
      2019 Research-status Report
  • [Presentation] 非線形積分が定める関数空間の完備性2019

    • Author(s)
      河邊 淳
    • Organizer
      RIMS共同研究(公開型)関数空間論とその周辺
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 非加法的測度が定める測度収束による可測関数空間の完備性2019

    • Author(s)
      山田直貴,河邊 淳
    • Organizer
      第8回信州関数解析シンポジウム
    • Related Report
      2019 Research-status Report
  • [Presentation] Choquet積分が定めるLp空間の完備性2019

    • Author(s)
      伊崎秀範,河邊 淳
    • Organizer
      第8回信州関数解析シンポジウム
    • Related Report
      2019 Research-status Report
  • [Presentation] The Vitali convergence theorem for distribution-based nonlinear integrals2018

    • Author(s)
      Jun Kawabe
    • Organizer
      Summer Symposium in Real Analysis XLII
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] A perturbation method in the study of convergence theorems of nonlinear integrals2018

    • Author(s)
      Jun Kawabe
    • Organizer
      The 6th Asian Conference on Nonlinear Analysis and Optimization
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A unified approach to convergence theorems of distribution-based nonlinear integrals2018

    • Author(s)
      河邊 淳
    • Organizer
      日本数学会2018年秋季総合分科会 実函数論分科会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 測度収束関数列の非線形積分のp次収束性2018

    • Author(s)
      河邊 淳
    • Organizer
      RIMS共同研究(公開型)関数空間の一般化とその周辺
    • Related Report
      2018 Research-status Report
  • [Presentation] 積分汎関数に対するVitali型収束定理2018

    • Author(s)
      河邊 淳
    • Organizer
      RIMS研究集会,関数空間の深化とその周辺
    • Related Report
      2017 Research-status Report
  • [Presentation] Convergence theorems of nonlinear integrals2017

    • Author(s)
      Jun Kawabe
    • Organizer
      The 10th Anniversary Conference on Nonlinear Analysis and Convex Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Convergence theorems of nonlinear integral functionals2017

    • Author(s)
      Jun Kawabe
    • Organizer
      Positivity IX
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] The Vitali convergence theorem for nonlinear integrals2017

    • Author(s)
      Jun Kawabe
    • Organizer
      6th International Eurasian Conference on Mathematical Sciences and Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 非線形積分の収束定理とその統合化2017

    • Author(s)
      河邊 淳
    • Organizer
      数理経済学会セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 非線形積分の収束定理の統一的定式化2017

    • Author(s)
      河邊 淳
    • Organizer
      第56回実函数論・函数解析学合同シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Choquet積分に対するVitaliの収束定理2017

    • Author(s)
      河邊 淳
    • Organizer
      日本数学会2017秋季総合分科会 実函数論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] 対称・反対称Choquet積分に対するVitaliの収束定理2017

    • Author(s)
      河邊 淳
    • Organizer
      実解析学シンポジウム2017
    • Related Report
      2017 Research-status Report
  • [Presentation] 一様Choquet可積分性とVitali型収束定理2017

    • Author(s)
      河邊 淳
    • Organizer
      第22回曖昧な気持ちに挑むワークショップ
    • Related Report
      2017 Research-status Report
  • [Remarks] 信州大学学術情報オンラインシステムSOAR

    • URL

      http://soar-rd.shinshu-u.ac.jp/profile/ja.jaAzZVkh.html

    • Related Report
      2021 Research-status Report
  • [Remarks] 信州大学学術情報オンラインシステムSOAR

    • URL

      http://soar-rd.shinshu-u.ac.jp/profile/ja.jaAaZVkh.html

    • Related Report
      2018 Research-status Report
  • [Remarks] 信州大学学術情報オンラインシステムSOAR

    • URL

      http://soar-rd.shinshu-u.ac.jp/profile/ja.jaAaZVkh.html?lng=ja&id=jaAaZVkh

    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2024-01-30  

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