Definition of relative (sequential) compactness

Question

In the book of Billingsley (Convergence of Prob. Measures) relative compactness is defined two times ($(S,d)$ is a metric space):

1. A family $\Pi$ of probability measures is relatively compact if every sequence of elements of $\Pi$ contains a weakly convergent subsequence.

2. A set $A\subset S$ is relatively compact if its closure is compact. (it can be shown that this is the case if and only if every sequence has a convergent subsequence)

What's the connection between them? I find it confusing since the first def. is for a space of probability measures, the second for set's of a metric space. Some authors use for the first def. the name "relatively sequentially compact". Is the condition of convergent subsequences more general, so that it can be generalized to spaces which are not metric spaces?

Answer 1

The space $\mathcal M_1(E)$ of probability measures on a metric space $E$ equipped with the topology of weak convergence is a topological space. So, the usual definition of relative compactness for topological spaces applies.

If $E$ is separable, then $\mathcal M_1(E)$ is metrizable via the Lévy-Prohorov metric.

In a metric space, relative compactness and relative sequentially compactness are equivalent.