TITLE: Tightness of Measures, Riesz Representation for locally compact spaces
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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