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

• 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
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• 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 / 摂動性 / 有界収束定理 / 単調収束定理 / 測度の弱収束 / 積分の収束定理

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.

Research Products

• [Journal Article] A unified approach to the monotone convergence theorem for nonlinear integrals (2016)
  • Author(s): Jun Kawabe
  • Journal Title: Fuzzy Sets Syst. Volume: 304 Pages: 1-19
  • DOI: 10.1016/j.fss.2016.06.013
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The monotone convergence theorems for nonlinear integrals on a topological space (2016)
  • Author(s): Jun Kawabe
  • Journal Title: Linear and Nonlinear Analysis Volume: 2 Pages: 281-300
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] 非加法的測度と非線形積分 (2016)
  • Author(s): 河邊 淳
  • Journal Title: 数学 Volume: 68 Pages: 266-292
  • Peer Reviewed / Open Access
• [Journal Article] 分布型非線形積分の単調収束定理の統一化 (2016)
  • Author(s): 河邊 淳
  • Journal Title: 第21回曖昧な気持ちに挑むワークショップ講演論文集 Volume: 21 Pages: 23-28
  • Peer Reviewed / Open Access
• [Journal Article] 非線形積分収束定理への摂動法による統一的アプローチ (2016)
  • Author(s): 河邊 淳
  • Journal Title: 実解析学シンポジウム2016講演論文集 Volume: 2016 Pages: 107-112
  • Open Access
• [Journal Article] Weak convergence of nonadditive measures based on nonlinear integral functionals (2016)
  • Author(s): Jun Kawabe
  • Journal Title: Fuzzy Sets and Systems Volume: 289 Pages: 1-15
  • DOI: 10.1016/j.fss.2015.02.011
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The structural characteristics of Choquet functionals (2015)
  • Author(s): Jun Kawabe
  • Journal Title: Journal of Nonlinear and Convex Analysis Volume: 16 Pages: 2181-2192
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Choquet積分による非加法的Portmanteau定理の定式化 (2015)
  • Author(s): 河邊 淳
  • Journal Title: 第20回曖昧な気持ちに挑むワークショップ講演論文集 Volume: 20 Pages: 55-57
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The bounded convergence in measure theorem for nonlinear integral functionals (2015)
  • Author(s): Jun Kawabe
  • Journal Title: Fuzzy Sets and Systems Volume: 271 Pages: 31-42
  • DOI: 10.1016/j.fss.2014.06.005
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Choquet汎関数とその表現測度における擬加法的性質の遺伝性 (2014)
  • Author(s): 河邊 淳
  • Journal Title: 第19回曖昧な気持ちに挑むワークショップ講演論文集 Volume: 19 Pages: 13-18
  • Peer Reviewed / Open Access
• [Presentation] Choquet積分のVitali型収束定理 (2017)
  • Author(s): 河邊 淳
  • Organizer: 京都大学数理解析研究所研究集会「関数空間の構造とその周辺」
  • Place of Presentation: 京都大学数理解析研究所，京都市，京都府
  • Year and Date: 2017-02-06
• [Presentation] 非線形積分に対する有界収束定理の定式化 (2016)
  • Author(s): 鈴木隆史，河邊 淳
  • Organizer: 第5回信州関数解析シンポジウム
  • Place of Presentation: 信州大学理学部，松本市，長野県
  • Year and Date: 2016-12-19
• [Presentation] 分布型非線形積分の単調収束定理の統一化 (2016)
  • Author(s): 河邊 淳
  • Organizer: 第21回曖昧な気持ちに挑むワークショップ (HM 2016)
  • Place of Presentation: コンパルホール，大分市，大分県
  • Year and Date: 2016-12-02
• [Presentation] 非線形積分収束定理への摂動法による統一的アプローチ (2016)
  • Author(s): 河邊 淳
  • Organizer: 実解析学シンポジウム2016
  • Place of Presentation: 奈良女子大学理学部，奈良市，奈良県
  • Year and Date: 2016-10-21
• [Presentation] 非線形積分の単調収束定理 (2016)
  • Author(s): 河邊 淳
  • Organizer: 日本数学会2016秋季総合分科会
  • Place of Presentation: 関西大学千里山キャンパス，吹田市，大阪府
  • Year and Date: 2016-09-15
• [Presentation] An approach to limit theorems for nonlinear integrals by perturbative method (2016)
  • Author(s): Jun Kawabe
  • Organizer: The Asian Mathematical Conference 2016 (AMC 2016)
  • Place of Presentation: Bali Nusa Dua Convention Center, Bali, Indonesia
  • Year and Date: 2016-07-25
  • Int'l Joint Research
• [Presentation] A perturbative approach to convergence theorems for nonlinear integrals (2016)
  • Author(s): Jun Kawabe
  • Organizer: International Symposium on Aggregation and Structures (ISAS 2016)
  • Place of Presentation: Hotel Parc Belle-Vue, Luxembourg city, Luxembourg
  • Year and Date: 2016-07-05
  • Int'l Joint Research
• [Presentation] A unified approach to limit theorems in nonlinear integration theory (2016)
  • Author(s): Jun Kawabe
  • Organizer: Summer Symposium in Real Analysis XL
  • Place of Presentation: International University of Sarajevo, Sarajevo, Bosnia and Herzegovina
  • Year and Date: 2016-06-19
  • Int'l Joint Research
• [Presentation] A unified approach to limit theorems for nonlinear integrals (2016)
  • Author(s): Jun Kawabe
  • Organizer: The Thirteenth International Conference on Fuzzy Set Theory and Applications
  • Place of Presentation: Hotel SOREA Maj, Liptovsky Jan (Slovak Republic)
  • Year and Date: 2016-01-24
  • Int'l Joint Research
• [Presentation] 摂動法による線形・非線形積分の極限定理への統一的アプローチ (2015)
  • Author(s): 河邊 淳
  • Organizer: 第24回信州数学セミナー
  • Place of Presentation: 信州大学理学部（長野県・松本市）
  • Year and Date: 2015-12-06
• [Presentation] 非加法的測度論への招待 (2015)
  • Author(s): 河邊 淳
  • Organizer: 第4回信州関数解析シンポジウム
  • Place of Presentation: 信州大学理学部（長野県・松本市）
  • Year and Date: 2015-12-03
  • Invited
• [Presentation] Choquet積分による非加法的Portmanteau定理の定式化 (2015)
  • Author(s): 河邊 淳
  • Organizer: 第20回曖昧な気持ちに挑むワークショップ
  • Place of Presentation: 明治大学中野キャンパス（東京都・中野区）
  • Year and Date: 2015-11-13
• [Presentation] Portmanteau定理の非加法化 (2015)
  • Author(s): 河邊 淳
  • Organizer: 実解析学シンポジウム2015
  • Place of Presentation: 東邦大学習志野キャンパス（千葉県・船橋市）
  • Year and Date: 2015-10-23
• [Presentation] 非加法的Portmanteau定理 (2015)
  • Author(s): 河邊 淳
  • Organizer: 日本数学会2015年秋季総合分科会実函数論分科会
  • Place of Presentation: 京都産業大学（京都府・京都市）
  • Year and Date: 2015-09-13
• [Presentation] Bounded convergence theorems for nonlinear integrals (2015)
  • Author(s): Jun Kawabe
  • Organizer: The Fifth International Symposium on Banach and Function Spaces 2015
  • Place of Presentation: 九州工業大学戸畑キャンパス（福岡県・北九州市）
  • Year and Date: 2015-09-02
  • Int'l Joint Research / Invited
• [Presentation] The weak topology of nonadditive measures based on nonlinear integral functionals (2015)
  • Author(s): Jun Kawabe
  • Organizer: The Thirty-Ninth Summer Symposium in Real Analysis
  • Place of Presentation: St. Olaf College, Northfield, MN (USA)
  • Year and Date: 2015-06-08
  • Int'l Joint Research
• [Presentation] 非加法的測度論における非線形積分汎関数の収束定理 (2015)
  • Author(s): 河邊 淳
  • Organizer: 関数解析学の研究とその応用
  • Place of Presentation: 新潟大学駅南キャンパス「ときめいと」（新潟県新潟市）
  • Year and Date: 2015-01-29 – 2015-01-30
• [Presentation] Sugeno積分の有界収束定理 (2014)
  • Author(s): 松本和也，河邊 淳
  • Organizer: 第3回信州関数解析シンポジウム
  • Place of Presentation: 信州大学理学部（長野県松本市）
  • Year and Date: 2014-12-01 – 2014-12-02
• [Presentation] Some convergence theorems of nonlinear integral functionals (2014)
  • Author(s): Jun Kawabe
  • Organizer: International Conference on Recent Advances in Pure and Applied Mathematics
  • Place of Presentation: Club Hotel Sera (Antalya, Turkey)
  • Year and Date: 2014-11-06 – 2014-11-09
• [Presentation] 非線形汎関数に対する有界収束定理 (2014)
  • Author(s): 河邊 淳
  • Organizer: 実解析学シンポジウム2014富山
  • Place of Presentation: 富山大学（富山県富山市）
  • Year and Date: 2014-10-31 – 2014-11-02
• [Presentation] Choquet汎関数とその表現測度における擬加法的性質の遺伝性 (2014)
  • Author(s): 河邊 淳
  • Organizer: 第19回曖昧な気持ちに挑むワークショップ
  • Place of Presentation: ホテル中原別荘（鹿児島県鹿児島市）
  • Year and Date: 2014-10-03 – 2014-10-04
• [Presentation] 分布関数で定まる非線形積分汎関数の有界収束定理 (2014)
  • Author(s): 河邊 淳
  • Organizer: 日本数学会2014年秋季総合分科会実函数論分科会
  • Place of Presentation: 広島大学東広島キャンパス（広島県東広島市）
  • Year and Date: 2014-09-25 – 2014-09-28
• [Presentation] Bounded convergence theorem for nonlinear integral functionals (2014)
  • Author(s): Jun Kawabe
  • Organizer: Summer Symposium in Real Analysis XXXVIII
  • Place of Presentation: Czech Technical University (Prague, Czech Republic)
  • Year and Date: 2014-07-07 – 2014-07-14
• [Remarks] 学術情報オンラインシステムSOAR
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• [Remarks] 研究者総覧
  • URL: http://soar-rd.shinshu-u.ac.jp/profile/ja.jaAaZVkh.html
• [Remarks] 研究者総覧 河邊 淳
  • URL: http://soar-rd.shinshu-u.ac.jp/profile/ja.jaAaZVkh.html?lng=ja&id=jaAaZVkh