Weak convergence of vector measures on topological spaces and its applications

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We have studied weak convergence of vector measures with values in Banach spaces and nuclear spaces, and have applied it to several interesting problems in real analysis, probability theory, control theory and so on. Some of our important results are as follows :1. We proved that the weak convergence of a net of tensor products of vector measures with values in nuclear spaces follows from that of the corresponding net of real product measures.2. We gave weak (sequential) compactness criteria for Radon vector measures with values in several locally convex spaces. Consequently, our result contains previous criteria obtained by Prokhorov-LeCam for real measures and by Marz and Shortt for Banach space valued measures.3. We extended a type of Strassen's theorem for probability measures to positive vector measures with values in the weak dual of a barreled locally convex space which has certain order conditions.4. We examined the convergence of feedback laws constructed by Mamdani method and the convergence of solutions of a non-linear state equation in fuzzy control system. As an application, we ensured the existence of fuzzy optimal control.5. We proved a theorem on the generalized Norlund summability (N, pn, qn) of Legendre series, which contains previous results by V.Singh, N.P.Mishra and Ajit Singh. It is also an Legendre analogue of a result due to Rajagopal.6. We proved some theorems on the generalized Norlund summability (N, pn, qn) of Fourier series and its conjugate series, which contain several previous results by A.N.Singh and S.P.Khare.