Q:

Tightness of Measures, Riesz Representation for locally compact spaces

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.

A:

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.

