Weak convergence of vector measures on topological spaces and its applications

我们研究了Banach空间和核空间中值向量测度的弱收敛问题，并将其应用于实分析、概率论、控制论等几个有趣的问题。我们的一些重要结果如下：1.证明了核空间中具有值的向量测度的张量积网弱收敛于相应的实积测度网的弱收敛。给出了取值于几个局部凸空间的Radon向量测度的弱(序列)紧性判据。因此，我们的结果包含了Prokhorov-LeCam关于实测度和Marz和Shortt对于Banach空间值测度得到的以前的准则。将一类关于概率测度的Strassen定理推广到具有一定序条件的桶形局部凸空间的弱对偶上的正向量测度。研究了用Mamdani方法构造的反馈律的收敛性和模糊控制系统中一类非线性状态方程解的收敛问题。作为应用，我们保证了模糊最优控制的存在性。证明了关于Legendre级数的广义Norlund可和(N，pn，qn)的一个定理，它包含了V.Singh，N.P.Mishra和Ajit Singh的前人的结果。这也是拉贾戈帕尔的一个结果的Legendre类比。证明了关于傅里叶级数及其共轭级数的广义Norlund可和(N，pn，qn)的几个定理，这些定理包含了A.N.Singh和S.P.Khare以前的几个结果。

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.