Borel概率空间的拓扑描述
Topological description of the Borel probability space
摘要：
本文研究了拓扑环境空间上Borel概率空间上几种常见拓扑的性质。我们证明了这两种流行的淡拓扑是相互等价的情况下，周围空间是LCH。由概率测度的集序贯收敛的两种等价描述导出的两类集拓扑也是等价的，与周围空间上的拓扑无关。给出了Borel概率空间上两类淡拓扑和两类集态拓扑可分或可度量化的明确条件。这些条件要么是Borel $σ$ -代数中基本事件的基数，要么是对周围空间的一些直接拓扑假设。在环境空间是紧的度量空间的情况下，给出了概率测度族是集相对紧的充要条件.最后给出了在Borel概率空间上构造新拓扑的一些推广问题和启发式方案。
英文摘要：
We study properties of some popular topology on the space of Borel probabilities on a topological ambient space in this paper. We show that the two types of popular vague topology are equivalent to each other in case the ambient space is LCH. The two types of setwise topology induced from two equivalent descriptions of setwisely sequential convergence of probability measures are also equivalent to each other regardless of the topology on the ambient space. We give explicit conditions for the two types of vague topology and the two types of setwise topology to be separable or metrizable on the space of Borel probabilities. These conditions are either in terms of the cardinality of the elementary events in the Borel $σ$-algebra or some direct topological assumptions on the ambient space. We give an necessary and sufficient condition for families of probability measures to be setwisely relatively compact in case the ambient space is a compact metric space. There are some extending problems and heuristic schemes on formulating new topologies on the space of Borel probabilities at the end of the work.
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