Weak convergence of positive vector measures with applications to real analysis

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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