"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets

Question

I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable.

In the discussion the set $X$ is always assumed to be a metric space, and let $\mathcal{P}(X)$ be the space of all Borel probability measures on $X$ equipped with weak convergence topology.

• In Billingsley's textbook, a family of Borel probability measures $M\subset\mathcal{P}(X)$ is said to be relatively compact if every sequence in $M$ has a convergent subsequence with limit in $\mathcal{P}(X)$.

Here we also have relative compactness and sequential compactness for general topological spaces:

• A set $A\subset X$ is relatively compact if $\bar{A}$ is compact.
• A set $A\subset X$ is sequentially compact if every sequence in $A$ has a convergent subsequence with limit in $A$.

Billingsley's relative compactness is different from the relative compactness in general topology (and also from sequential compactness), so I cannot see why we say such families of measures are relatively compact. So far, I have found a related question, which assumes $X$ is a polish space.

https://math.stackexchange.com/questions/3640221/prokhorovs-theorem-the-statement-precompact-sequentially-compact-relativel

In that question user87690 argued that the term "relatively sequentially compact" is more appropriate. Also since in that question $X$ was assumed to be a polish space, sequential compactness is equivalent to compactness so that we can simply say "relatively compact". One problem for me is that $\mathcal{P}(X)$ is not always metrizable, although it is metrizable if $X$ is separable as polish spaces are.

So I may guess that, as user87690 suggested, "relatively compact" stands for "relatively sequentially compact", and one shortened the term because either we usually deal with metric spaces $X$ which are at least separable, or simply "relatively sequentially compact" is too long. But I'm still looking for more persuasive explanations.

Any answers will be appreciated. Thank you!

Answer 1

Only a partial answer. An interesting result in this context is V.I. Bogachev, Measure Theory, Vol. 2, (2006), Th. 8.3.2. which in particular says that the set $\cal{P}_\tau(X)$ of all $\tau$-additive probability measures on $X$ is metrizable with the Levy-Prokhorov metric. If $\cal{P}_\sigma(X) \not= \cal{P}_\tau(X)$, $\cal{P}_\sigma(X)$ the set of $\sigma$-additve probability measures on $X$, then the weak topology is not metrizable. Further note that $\cal{P}_\sigma(X) = \cal{P}_\tau(X)$ is consistent in ZFC. So in this case both compactnes-criteria are equivalent. I don't know if it is consistent that both compactnes-criteria are not equivalent.