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?

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?