论文标题
kantorovich类型的拓扑结合了衡量标准的拓扑
Kantorovich type topologies on spaces of measures and convergence of barycenters
论文作者
论文摘要
我们研究了两个拓扑$τ_{kr} $和$τ_k$在措施的空间上，这是由Kantorovich--Rubinshtein和Kantorovich Eminorms在公制空间中类似于其经典规范的完全规则空间上。 Kantorovich-Rubinshtein拓扑$τ_{kr} $与非负措施和有限的统一紧密措施集的弱拓扑相吻合。对于Kantorovich拓扑的紧凑性，给出了足够的条件。我们表明，对于对数凹入量和稳定的度量，弱收敛意味着坎多洛维奇拓扑结构的收敛性。我们还获得了有效验证的条件，用于从序列或净局部在局部凸出空间上弱收敛的ra量测量的条件。作为一种应用，表明对于弱收敛性地对数凹形测量和稳定的措施，其重中心的融合在没有其他条件的情况下保持。相对于对数凹入度量，固定程度的多项式密度给出的措施也是如此。
We study two topologies $τ_{KR}$ and $τ_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $τ_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.
