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

1.对于两个Banach空间值向量测度，证明了Borel内射张量积的存在唯一性和一个fubini型定理的有效性。基于这一结果，将拓扑半群上Banach代数值向量测度的卷积定义为它们的Borel内射张量积和半群运算所引起的测度。证明了向量测度的Borel内射张量积或卷积的联合弱连续性。证明了某些Banach格中正向量测度的内射张量积对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的双线性向量积分理论。3 .给出了Frechet空间、半自反空间和半montel空间中具有值的向量测度的Prokhorov-LeCam紧性准则和Varadarajan度量性准则。给出了完全可分度量空间上的一组向量测度的紧性和序紧性判据。其中，证明了当且仅当对应的实测度集相对于测度的通常弱收敛性是相对序紧的，这样的向量测度集是一致有界和一致紧的。在Dedekind完备Riesz空间值σ测度中证明了Portmanteau定理的有效性。从结果可以认识到，不是范数，而是向量测度取值的空间上的顺序结构对Portmanteau定理的有效性至关重要。对于完全正则空间上的Dedekind完备Riesz空间值σ-测度，证明了Borel积的存在唯一性。证明了Dedekind完备Riesz空间值σ-测度网在一致有界、一致等连续函数类上的弱阶收敛性是一致的。证明了在弱σ-分布的Dedekind完备Riesz空间中，任何完备可分度量空间上的每一个Borel σ-测度都是自动有序紧的。少

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