**Problem set up:**

Consider $C_b$, the Banach space of continuous bounded functions on $[0, \infty)$ equipped with the sup norm. Denote by $M$ the set of probability measures on $[0, \infty)$, and for $r > 0$ denote by $M_r$ the set of probability measures supported on $[r, \infty)$. We will consider $M$ as subsets of the continuous dual $C_b^*$ in the usual way.

Consider the set $\mathcal S$ of linear functionals $L \in C_b^*$ such that there exist a sequence $r_n$ of real numbers with $r_n \to \infty$, and a sequence of probability measures $\mu_n$ with $\mu_n \in M_{r_n}$ for all $n$ such that $\mu_n \to L$ in the weak* topology.

By the Banach-Alaoglu theorem, $\mathcal S$ is nonempty.

Question:Is it possible to produce an explicit example of an element of $\mathcal S$, and the corresponding probability measures?

sequentiallyclosed in the weak* topology of $C_b^{\ast}$, so $\mathcal{S}$ would be empty, if you didn't allow nets. I don't have a reference, though, so take it with a grain a salt. $\endgroup$6more comments