Refinement of convergence theorems of nonlinear integrals with applications to the topological properties of function spaces determined by nonlinear integrals

非线性积分收敛定理的细化及其对非线性积分确定的函数空间拓扑性质的应用

• 批准号: 20K03695
• 负责人: 河邊 淳
• 金额: $ 2.75万
• 依托单位: Shinshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2020
• 资助国家: 日本
• 起止时间: 2020-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20K03695/
• 关键词: 非加法的測度; Choquet積分; Sugeno積分; Shilkret積分; Lorentz空間; 性質(C); 完備性; 可分性; 可測関数空間; 開球条件; 上からの自己連続性; 疑距離生成性; 零加法性; 順序連続性; 非線形積分; 積分の収束定理; 関数空間; ショケ積分

１．実数値可測関数全体からなる線形空間上には，非加法的測度μを用いて定義されたSugeno積分型，Choquet積分型，Ky-Fan型，Dunford-Schwartz型の球（ball）の他に，AssaとZimperが導入した距離関数を用いない球から導入された５つの位相が提案されている．この研究では，これら５つの位相を比較し，次の結果を得た．(a)μが有限ならば，これら５つの位相はすべて一致する．(b)これらの位相が第１可算公理を満たし，位相に関する収束とμ-測度収束とが同値となるための必要十分条件は，球が位相の基本近傍系となることである．(c)μが上から連続かつ上から自己連続ならば，AssaとZimperが導入した球は常に開集合となる．２．非加法的測度μによるSugeno積分を用いて定義したLorentz空間（以下ではSL空間とよぶ）の性質を調べ，次の結果を得た．(a)μが擬距離生成的かつ下から単調自己連続で性質(C)をもてば，SL空間は完備である．(b)μが擬距離生成的かつ順序連続で可算基をもてば，SL空間は可分である．(c)Lorentz型の擬距離が定める位相（以下ではSL位相という）に関して，開球が開集合となるための必要十分条件はμの上からの自己連続性，閉球が閉集合となるための必要十分条件はμの下からの自己連続性である．(d)SL位相はLorentz型の関数空間を定める際の指数によらずただ一つに定まる．(e)SL空間は，μが上から自己連続ならばT1空間，μが上から自己連続かつ擬距離生成的ならばHausdorff空間，μが自己連続ならば正則空間，μが反対称緩劣加法的ならば正規空間となる．(f)μが劣加法的ならば，SL空間は距離付け可能である．(g)μが順序連続かつ上から自己連続で緩劣加法的ならば，SL空間は開球条件を満たす線形位相空間である．

1. It can be counted as a whole. The non-additive measurement μ is defined by the definition of Sugeno positive classification, Choquet positive classification, Ky- Fan type, Dunford- Schwartz ball (ball) filter, and Assa Zimper entry. The distance separation number is loaded into the cluster 5-inch phase measurement proposal using the drop-off ball, and the phase comparison between the two is used. The results of the second test show that: (a) the phase of the first phase is equal to that of the axiom, and the phase of the first phase is equal to that of the axiom. (a) the results of the secondary results show that. (a) the results show that. (a) the results of the secondary results show that. (a) the phase of the first phase is equal to that of the axiom. (B) the phase of the phase is similar to that of the beam. the results show that. (a) the results of the secondary results show that. (a) the results show that: (a) the phase is limited and the phase is consistent. (B) the phase of phase 1 can be calculated as axiomatic. The non-additive measurement of the non-additive Sugeno is used to define the Lorentz space (the following SL space). The secondary results are satisfactory. (a) under the cloud generated by the remote distance separation, you can connect to your own connection properties (C). The SL space device is ready to operate. (B) the serial link generated by the remote distance separation can be counted as the base device, and the SL space device can be separated. (C) the Lorentz distance detector phase (the following SL phase) is sensitive, and it is necessary to open the ball to the assembly device. It is necessary to set the device set to connect with each other. (d) SL phase Lorentz-type connection index. (e) SL space, μ-switch self-connection connection T1 space, μ-port self-connection connection distance generated customer Hausdorff space, and μ user self-connection filter policy space. (F) the error of the bad addition. (F) the bad addition of the SL space distance may be different. (G) on the μ serial link link, you can link the bad addition of your own, and the SL space opening condition does not match the shape and phase of the line.

项目成果

非加法的測度が定めるLorentz空間の完備性

由非相加测度确定的洛伦兹空间的完备性

• 发表时间: 2023
• 作者: Jun Kawabe;野邊厚;R. Kajikiya and E. Ko;Michinori Ishiwata;Jun Kawabe;河邊 淳
• 通讯作者: 河邊 淳

The completeness and separability of function spaces in nonadditive measure theory

• DOI: 10.1016/j.fss.2022.10.001
• 发表时间: 2022-06
• 期刊: Fuzzy Sets Syst.
• 作者: J. Kawabe;N. Yamada

The completeness and separability of the Lorentz spaces defined by the Sugeno and Shilkret integrals

Sugeno 和 Shilkret 积分定义的洛伦兹空间的完备性和可分性

• 发表时间: 2021
• 期刊: Linear and Nonlinear Analysis
• 作者: Jun Kawabe;野邊厚;R. Kajikiya and E. Ko;Michinori Ishiwata;Jun Kawabe
• 通讯作者: Jun Kawabe

信州大学学術情報オンラインシステム SOAR

信州大学学术信息在线系统SOAR

非加法的測度空間上の可測関数空間の線形位相構造

非可加测度空间上可测函数空间的线性拓扑结构

• 发表时间: 2022
• 期刊: 実解析学シンポジウム講演論文集
• 作者: Ishiwata;Michinori; Ruf;Bernhard; Sani;Federica; Terraneo;Elide;村田 実貴生;河邊 淳
• 通讯作者: 河邊 淳