Weak convergence of vector measures with applications to real analysis

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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, differential geometry and so on. Some of our important results are as follows :1. By an essential use of Bartle's bilinear integration theory, 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.2. The weak compactness of a set of control inputs is shown in the case that they are given by the gravity calculation of time dependent fuzzy membership functions. As an application, the existence of optimal solution is discussed in a fuzzy control for an open-loop system.3. We obtain a convergence theorem of compound probability measures on a uniform space for a net of uniformly equicontinuous transition probabilities. This result applies to Gaussian transition probabilities on a Hilbert spaces.4. We obtain a general theorem for the method (N,p_n, q_n)(C, 1) summability of the sequence {nB_n (x)}, which contains some theorems due to S.P.Khare, V.K.Tripathi and A.N.Singh and et al.5. It is shown that some central manifold exists in a neighborhood of a point of equilibrium.6. It is shown that a set of fuzzy membership functions in the NBV space is compact with respect to the weak^* topology. This result applies to the existence of fuzzy optimal control.7. A relation between two convergence theorems of maritingale in the limit, i.e., L^1-boundedness and integrability of stopped processes is studied.8. We define another almost complex structure (resp.almost contact structure) and an indefinite Kahlerian (resp. Sasakian) manifold with affine connection.