Weak convergence of positive vector measures with applications to real analysis

正向量测量与实际分析应用的收敛性较弱

• 批准号: 13640162
• 负责人: KAWABE Jun
• 金额: $ 2.05万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640162/
• 关键词: weak convergence; tensor product; injective tensor integral; Fourier coefficient; compactness; martingale-like sequence; H^∞ control problem; semi-Riemannian manifold; ジョルダン多角形; 一様緊密性; H^∞制御問題; アフィン接続

We have studied weak convergence of positive vector measures with values in Banach spaces and nuclear spaces, and have applied to several interesting problems in real analysis, probability theory, control theory, differential geometry and so on. Some of our important results are as follows:1. A sequential compactness criterion is given for the weak topology of vector measures with values in certain nuclear spaces.2. It is shown that the injective tensor product of positive vector measures in certain Banach lattices is jointly continuous with respect to the weak convergence of vector measures. Our approach to this problem is based on Bartle's bilinear integration theory.3. Prokhorov-LeCam' s compactness criteria and Varadarajan' s metrizability criterion are given for vector measures with values in Frechet spaces, semi-reflexive spaces, and semi-Montel spaces.4. The existence and uniqueness of the Borel injective tensor product of two Banach space-valued vector measures and the validity … More of a Fubini-type theorem are shown. Thanks to these results, the convolution of vector measures on a topological semigroup is defined as the measure induced by their Borel injective tensor product and the semigroup operation. The joint weak continuity of Borel injective tensor products or convolutions of vector measures is also proved.5. Compactness and sequential compactness criteria are given for a set of vector measures on a complete separable metric space with values in a certain semi-Montel space.6. It is shown that the Portmanteau Theorem remains valid for order σ-additive, positive vector measures with values in a Dedekind σ-complete Riesz space.7. A general theorem for the method of absolute Norulund summability is given.8. Some new characterizations of compact sets are given.9. The uniqueness of the solutions in H^∞ control problems is proved for general plants. Another proof, which does not depend on the complexity of the form of the algebraic Riccati equations, is also given.10. It is proved that there is a qubic-free sequence of letters.11. It is shown that a conformal Killing vector field is parallel on a compact almost Kahlerian manifold. Less

本文研究了取值于Banach空间和核空间的正向量测度的弱收敛性，并应用于真实的分析、概率论、控制论、微分几何等领域中的一些有趣的问题，主要结果如下：1.给出了取值于核空间的向量测度的弱拓扑的序列紧性准则.证明了某些Banach格上正向量测度的内射张量积关于向量测度的弱收敛是联合连续的。我们解决这个问题的方法是基于Bartle的双线性积分理论。对于取值于Frechet空间、半自反空间和半Montel空间的向量测度，给出了Prokhorov-LeCam紧性准则和Varadarajan度量化准则.两个Banach空间值向量测度的Borel内射张量积的存在唯一性及其有效性 ...更多信息 示出了富比尼型定理的。利用这些结果，定义了拓扑半群上向量测度的卷积为它们的Borel内射张量积和半群运算所诱导的测度。证明了向量测度的Borel内射张量积或卷积的联合弱连续性.给出了完备可分度量空间上值在某个半Montel空间上的向量测度集的紧性和序列紧性准则.证明了Portmanteau定理对于值在Dedekind σ-完备Riesz空间中的阶σ-可加正向量测度仍然成立.给出了绝对Norulund可和性方法的一般定理.给出了紧集的一些新的刻画.对于一般对象，证明了H^∞控制问题解的唯一性.给出了不依赖于代数Riccati方程形式复杂性的另一个证明。它被证明是有一个qubic免费序列的字母。11.证明了紧致几乎Kahlerian流形上的共形Killing向量场是平行的。少

项目成果

J.Kawabe: "Sequential compactness for the weak topology of vector measures in certain nuclear spaces"Georgian Math.J.. 8. 283-295 (2001)

J.Kawabe：“某些核空间中矢量测度弱拓扑的顺序紧性”Georgian Math.J.. 8. 283-295 (2001)

J.Kawabe: "Joint continuity of injective tensor products of vector measures in Banach lattices"J.Aust.Math.Soc.. 73. 1-15 (2002)

J.Kawabe：“Banach 格子中矢量测度的单射张量积的联合连续性”J.Aust.Math.Soc.. 73. 1-15 (2002)

J.Kawabe: "Strassen's theorem for positive vector measures"京都大学数理解析研究所講究録. 1298. 59-64 (2002)

J.Kawabe：“正向量测度的斯特拉森定理”京都大学数学科学研究所 Kokyuroku。1298. 59-64 (2002)。

遠藤 登: "時間依存するファジィ集合族を用いた最適制御問題に関する一考察"京都大学数理解析研究所講究録. 1186. 216-223 (2001)

Noboru Endo：“使用时间相关模糊集族的最优控制问题的研究”京都大学数学科学研究所 Kokyuroku。1186. 216-223 (2001)。

Jun Kawabe: "Weak convergence of injective tensor products of vector measures"AMC 2000 Proceedings. (印刷中).

Jun Kawabe：“矢量测量的单射张量积的弱收敛”AMC 2000 论文集（正在出版）。