Topological description of the Borel probability space.

Given a topological ambient space $X$, We study properties of some popular topology on the Borel probability space $\mathcal{M}(X)$ in this paper. We show that the two types of vague topology are equivalent to each other in case the ambient space $X$ is LCH. The two types of setwise topology induced from two equivalent descriptions of setwise convergence of sequences of probability measures are also equivalent to each other for any ambient space $X$. We give explicit conditions for the two types of vague topology and the two types of setwise topology to be separable or metrizable on $\mathcal{M}(X)$. These conditions are either in terms of the cardinality of the elementary events in the Borel $\sigma$-algebra $\mathscr{B}$ or some direct topological assumptions on the ambient space $X$. We give the necessary and sufficient condition for families of probability measures to be setwisely relatively compact in case $X$ is a compact metric space. There are some extending questions on the topology of the Borel probability space $\mathcal{M}(X)$ at the end of the work.