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Weak order convergence of Riesz space-valued positive vector measures with applications

Riesz空间值正向量测度的弱阶收敛及其应用

基本信息

批准号：
15540162
负责人：
KAWABE Jun
金额：
$ 2.24万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540162/
关键词：
Riesz space-valued measures weak convergence of measures Riesz type representation theorem portmanteau theorem bilinear vector integration Borel product of vector measures compactness criterion uniform order tightness リース空間値測度ベクトル測度のボレル

项目摘要

1.The existence and uniqueness of the Borel injective tensor product and the validity of a Fubini-type theorem are shown for two Banach space-valued vector measures. Thanks to this result, the convolution of Banach algebra-valued 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.2.It is shown that the injective tensor product of positive vector measures in certain Banach lattices is jointly continuous with respect to weak convergence of vector measures. Our approach to this problem is based on Bartle's bilinear vector 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.Compactness and sequential compactness criterion are given … More for a set of vector measures on a complete separable metric space with values in a certain semi-Montel space. Among others it is shown that a set of such vector measures is uniformly bounded and uniformly tight if and only if the corresponding set of real measures is relatively sequentially compact with respect to the usual weak convergence of measures.5.The validity of a version of the Portmanteau Theorem is shown for Dedekind complete Riesz space-valued σ-measures. From the result one can recognize that not the norm but the order structure on the space where vector measures take values is essential to the validity of the Portmanteau Theorem.6.The existence and uniqueness of Borel products are proved for Dedekind complete Riesz space-valued σ-measures on completely regular spaces.7.It is shown that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions.8.It is shown that every Borel σ-measure on any complete separable metric space is automatically orderly tight in the case that the measure takes values in a weakly σ-distributive, Dedekind complete Riesz space. Less

1.对于两个Banach空间值向量测度，证明了Borel单射张量积的存在性和唯一性以及Fubini型定理的有效性。由于这个结果，拓扑半群上的 Banach 代数值向量测度的卷积被定义为由其 Borel 单射张量积和半群运算引起的测度。还证明了Borel单射张量积或向量测度卷积的联合弱连续性。2.证明了某些Banach格中正向量测度的单射张量积对于向量测度弱收敛而言是联合连续的。我们解决这个问题的方法基于 Bartle 的双线性向量积分理论。 3. 对于具有 Frechet 空间、半自反空间和半 Montel 空间中的值的向量测度，给出了 Prokhorov-LeCam 的紧致性准则和 Varadarajan 的可度量性准则。 4. 给出了完全可分离上的一组向量测度的紧致性和顺序紧致性准则。具有某个半蒙特尔空间中的值的度量空间。除其他外，它表明，当且仅当相应的实测度集相对于通常的弱收敛测度而言是相对顺序紧凑时，一组此类向量测度是一致有界和一致紧的。 5. 对于 Dedekind 完整的 Riesz 空间值 σ 测度，显示了 Portmanteau 定理版本的有效性。从结果可以看出，对于Portmanteau定理的有效性而言，向量测度取值空间上的阶结构不是范数。6.在完全正则空间上证明了Dedekind完全Riesz空间值σ测度的Borel积的存在性和唯一性。7.证明了Dedekind完全Riesz空间值σ测度网络的弱阶收敛性在完全正则空间上是一致的。一致有界、一致等连续的函数类。8.证明了任何完全可分度量空间上的每个 Borel σ 测度在弱 σ 分布、Dedekind 完全 Riesz 空间中取值的情况下都是自动有序紧的。较少的