Explicit example of a certain weak-* limit

Question

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?

Answer 1

(reading "sequence" as "net", as suggested in the comments)

Well, $C_b(\mathbb{R}^+) \cong C(\beta\mathbb{R}^+)$, so any such $L$ will arise from a probability measure on the Stone-Cech remainder $\beta\mathbb{R}^+ \setminus \mathbb{R}^+$. You need some choice principle to know this set is nonempty, so no example can be very explicit. (I think it's consistent with ZF that there are no free ultrafilters on $\mathbb{R}^+$.)

If you're willing to take ultrafilters as "explicit", then evaluation on a free ultrafilter on $\mathbb{Z}^+$ would be an example.