What is the topological dual of $C_b(\mathbb{R})$ Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm. 
1) Do we know a precise description of its topological dual $C_b(\mathbb{R})^*$ ? 
2) I was wondering what kind of relation has $C_b(\mathbb{R})^*$ with $C_c(\mathbb{R})^*$, the topological dual of the space $C_c(\mathbb{R})$ of continuous functions having compact support. 
If $L\in C_b(\mathbb{R})^*$ then  there exists $C_L$ such that 
$|L(f)|\leq C_L\|f\|_\infty$ for any $f\in C_b(\mathbb{R})$, and thus for all $f\in C_c(\mathbb{R})$, so that $L\in C_c(\mathbb{R})^*$ and $ C_b(\mathbb{R})^*\subset C_c(\mathbb{R})^*$. Is that correct ? 
Then, can we find all the probability measures on $\mathbb{R}$ into $C_b(\mathbb{R})^*$ ? 
(In other worlds, is the "weak" topology for probability measures a weak* one ?)
 A: The space $C^b(\mathbb R)$ is a commutative C*-algebra, hence by the Gelfand-Naimark theorem it is *-isomorphic to some $C(K)$ space ($K$ is the spectrum of this algebra and the *-isomorphism is just the Gelfand transform). The dual of a $C(K)$ space is described by the Riesz-Kakutani theorem as the space of all regular Borel measures on $K$.
EDIT: In this case, the spectrum of $C^b(\mathbb{R})$ is just $\beta(\mathbb{R})$ which can be seen directly using basic properties of the Gelfand transform.
A: Take the algebra $\mathcal A$ generated by the closed sets in $\mathbb R$.  The space of finitely-additive signed measures on $\mathcal A$, with variation norm, is the dual of $C_b(\mathbb R)$.
The top reference for this and many other interesting topics: Gillman & Jerison, Rings of Continuous Functions.  Note $\mathbb R$ is metrizable, so the "zero sets" in the book are just the closed sets.
Another model for the dual: Let $\beta\mathbb R$ be the Stone-Cech compactification of $\mathbb R$, so that $C_b(\mathbb R)$ is naturally identified with $C(\beta \mathbb R)$.  Then utilize your knowledge for the dual of $C(K)$ where $K$ is compact Hausdorff.
