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

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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