Tightness and Functional Analysis

Question

Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an element of the unit ball of $C_0(\mathbb{R})^*$. By Banach-Alaoglu, the unit ball is weak-$^*$ compact and since $C_0(\mathbb{R})$ is separable, we have that every sequence of probability measures has a weak-$^*$ limit. This limit might not be a probability measure, but is if the sequence is tight. My first question is there a nice functional analytic viewpoint of tightness, in this scenario?

In applications, it is important to extend the action of $\mu$ to all of $C_B(\mathbb{R})$ (since, for instance, that's where the characters are). Now $\mu$ no longer lives in the separable dual space of $C_0(\mathbb{R})$, but in the dual of $C_b(\mathbb{R})$ which is not separable. Thus Banach-Alaoglu no longer implies anything about sequences. In probability texts, they use very specific Lebesgue theory arguments to assert that if $\mu_n$ is a sequence of tight measures that there is a convergence sub sequence. My second question is in this more general setting, can we still find a functional analytic view point that will allow us to see this?

Answer 1

The functional analysis setting for this was established by R.C. Buck in the $1950$'s. He introduced a natural complete locally convex topology on $C^b(S)$, the space of bounded, continuous functionss on a locally compact space S$, the so- called strict topology with the properties

1) the dual space is the space of tight (Radon) measures on $S$;

2) if $ S$ is $\sigma$ compact, then a set of measures is weakly compact (for this duality) if and only if it is tight.

This was later extended to the case of general completely regular spaces and the question of which $S$ display the second property (in particular, with respect to families of probability measures) attracted much attention.