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Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, assume that $\mathfrak{C} = C_b(X)$, i.e. all bounded continuous functions on $X$).

Then, if $\mu_n(f), n \geq 1$ is Cauchy for all $f \in \mathfrak{C}$, does it follow that $\mu_n$ converges weakly?

Alternatively, if $C = C_b(X)$ then writing $m(f) = \lim_n \mu_n(f)$ does the Riesz Representation theorem apply to $m(f)$? In general, we need some tightness conditions on a functional $m(f)$ to apply RRT. However, given the particular form of the functional, is there a way to detect whether the sequence "loses mass at infinity"?

If not, is there a constructive counter-example of this?

Edit: OP here. Thanks for all the suggestions so far. To clarify something: I'm well aware of tightness of the sequence measures being equivalent to precompactness (i.e. Prokhorov's Theorem) which then through sequential compactness would imply a weak limit of $\mu_n$. The issue here is (or was) how to prove that the sequence is uniformly tight in the first place, given only that the integrals against functions $\mu_n(f)$ themselves converge.

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As you point out, the easy part is that uniform tightness of a family implies relative weak compactness since the firmer prop erty is just equicontinuity for the strict topology. For this reason, the converse property has been the object of a great deal of research, which is the reason I suggested looking at Conway for starters. Lookong through the literature is something you are going to have to do for yourself. –  berlin Mar 28 at 23:29

2 Answers 2

The proper framework for your question is the so-called strict topology on the space of bounded continuous functions which was introduced for the case of a locally compact space by R.C. Buck in the 50's and extended to the case of a completely regular space by several authors around 1970. It has the property that the dual is the space of tight Radon measures. In your case (locally compact and metric, and so paracompact), there are very strong results available, particularly in the direction that you are interested in---characterisations of compactness and convergence for families of measures. A central result concerns the relationship between weak compactness and uniform tightness of such families. You might start by consulting the article "The strict topology and compactness in the space of measures" by John B. Conway which appeared in the Transactions 126 (1967) 474-486. You could also look up the topic of Prohorov's theorem which is relevant to your query.

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I'm not sure that this question is using "tight" in the same sense that you are. –  Nate Eldredge Mar 28 at 0:47
    
@Nate Eldredge. To my knowledge, there is only one usage of tightness in this context and it coincides with being Radon, i.e., that for each positive $\epsilon$ there is a compact $K$ so that the measure of its complement is $\leq \epsilon$. –  barcelos Mar 28 at 6:21

A (specific and historical) complement to barcelos answer, too long to be correctly formatted as comment:

From Bourbaki's historical notes to chaper IX of integration:

A. D. Alexandroff [...] introduces a hierarchy in the set of positive linear forms on the space $C^b(X)$ of bounded continuous functions on a completely regular space $X$, he defines tight convergence of bounded measures and proves among others the following two theorems:

a) if $X$ is Polish, the set of linear forms on $C^b(X)$ corresponding to the measures is closed for the weak convergence of sequences;

b) if a sequence of bounded measures has a tight limit, "no mass escapes at infinity" (this is a weak form of the converse of Prokhorov's theorem on tight convergence).

Alexandroff's papers are online (in english and in russian), see Riesz's representation theorem for non-locally compact spaces

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