Prokhorov's Theorem : The Statement. Precompact, Sequentially Compact, Relatively Compact : Definitions.

Let $$S$$ be a Polish space (a complete separable metric space), Let $$\mathcal{P}(S)$$ be the space of Borel probability measures on $$S$$ (the Borel sets are induced by the metric on the space $$S$$). Next I give the statement of Prokhorov's Theorem as I know it.

$$\textbf{Prokhorov's Theorem :}$$ $$\textbf{ (Bilingsley Convergence of Probability Measures page 37) }$$ A subset $$\mathcal{M}\subset \mathcal{P}(S)$$ is tight if and only if it is relatively compact.

Now some definitions :

• $$\textbf{Definition 1. : (Bilingsley page 35)}$$ A subset $$\mathcal{M}\subset\mathcal{P}(S)$$ is $$\textit{relatively compact}$$ if every sequence has a weakly convergent sub-sequence. That is for $$\{\mu_k\}\subset \mathcal{M}$$, there exists $$\{\mu_{k_{m}}\}\subset \mathcal{M}$$ such that $$\mu_{k_{m}}\overset{weakly}{\longrightarrow}\mu \in \mathcal{P}(S)$$.

• $$\textbf{Definition 2. : (Wikipedia Webpage Sequentially Compact) }$$ A topological space $$X$$ is $$\textit{Sequentially Compact}$$ if every sequence in $$X$$ has a subsequence converging in $$X$$. (Notice the convergence is w.r.t the topology, not necessarily the weak topology.

• $$\textbf{Definition 3. : (Wikipedia Webpage Precompact) }$$ A subset $$Y$$ of a topological space $$X$$ is precompact (relatively compact) if its closure is compact.

$$\textbf{Question 1:}$$ The definitions 1. and 3. both use the term 'relatively compact' to mean different things, are either of them 'correct'?.

$$\textbf{Question 2:}$$ I have seen Prokhorov's Theorem stated slightly differently in some places : for example if a subset $$\mathcal{M}\subset \mathcal{P}(S)$$ is tight :

• Wikipedia says $$\mathcal{M}$$ is sequentially compact.

• In other places I have seen it said that $$\mathcal{M}$$ is precompact.

• Or we have that $$\mathcal{M}$$ is relatively compact, as I stated before.

Which of these is correct, are they in fact the same?

• Without going too far, the definitions of compact set are different in France and in the USA. Commented Apr 23, 2020 at 16:35
• For metric spaces, definitions $1$ and $3$ are equivalent. And Wikipedia states that the closure of $\mathcal M$ is sequentially compact which is equivalent to $\mathcal M$ being relatively compact. (Can you point out the difference between sequentially compact and relatively compact? Hint: look at the limits.)
– cqfd
Commented Apr 23, 2020 at 16:49

There are several issues.

1) There are different definitions of compactness – the “true” compactness meaning that every open cover has a finite subcover, and the sequential compactness meaning that every sequence has a convergent sequence. These are absolute properties of topological spaces – a subset is compact if it is compact with respect to the subspace topology as a space on its own. For metrizable spaces, compactness and sequential compactness are equivalent, and the space of Borel probability measures of a separable metrizable space is itself separable metrizable (and Polish iff the original space is Polish, and compact iff the original space is compact).

2) Relative compactness is a property of a subset of a topological space: a subset is relatively compact if its closure is compact (with respect to any definition of compactness considered). So the Definition 1 corresponds to being relatively sequentially compact with respect to the weak topology (which is one particular topology considered on $$P(S)$$). But there is no problem calling it just relatively compact (again, with respect to the weak topology) since the space is metrizable.

3) Precompactness has more meanings. In this case, it is just a synonym for relative compactness. But there is also a closely related meaning that is an absolute property: a metric (or more generally uniform) space is precompact if its completion is compact. This other meaning is also called or is equivalent to being totally bounded. The sequential variant of this absolute precompactness means that every sequence has a Cauchy subsequence. Note that the two meanings of precompactnes – relative compactness and total boundedness – are not equivalent: a subset that is totally bounded as a space on its own may have closure that is not compact, but the other implication holds.

This should settle your questions. All the statements are correct and the different definitions are consistent. Note that Wikipedia doesn't say that a tight set is sequentially compact, but that its closure is sequentially compact. I also encourage you to think about the definition between sequential compactness and relative sequential compactness, as suggested by @Thomas Shelby.