Weak convergence of positive vector measures with applications to real analysis

我们研究了Banach空间和核空间中取值的正向量测度的弱收敛问题，并应用于实分析、概率论、控制论、微分几何等领域中的几个有趣问题。我们的一些重要结果如下：1.给出了取值于某些核空间的向量测度的弱拓扑序列紧性的一个判据。证明了某些Banach格中正向量测度的内射张量积关于向量测度的弱收敛是联合连续的。我们解决这个问题的方法是基于Bartle的双线性积分理论。给出了Frechet空间、半自反空间和半Montel空间中具有值的向量测度的Prokhorov-LeCam的S紧性判据和Varadarajan的S的可度量化判据。两个Banach空间值向量测度的Borel内射张量积的存在唯一性及有效性…给出了更多的Fubini型定理。利用这些结果，将拓扑半群上的向量测度的卷积定义为它们的Borel内射张量乘积和半群运算所诱导的测度。证明了向量测度的Borel内射张量积或卷积的联合弱连续性。给出了取值于某个半Montel空间的完备可分度量空间上的一组向量测度的紧性和序列紧性准则。证明了混合定理对于值在σ-完全Riesz空间中的阶σ-可加正向量测度仍然有效。给出了绝对Norulund可和方法的一个一般定理。给出了紧集的一些新刻画。对于一般对象，证明了H^∞控制问题解的唯一性。还给出了不依赖于代数Riccati方程形式复杂性的另一种证明。证明了存在一个无二次方的字母序列。证明了在紧致的几乎Kahlerian流形上，共形Kill向量场是平行的。较少

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