Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz representation theorem) that the continuous dual of $C_c$ can be identified the vector space of all finite signed Radon measures on $\mathbb{R}$, that is set-functions $\mu : \mathscr{A} \to \mathbb{R}$ (that are regular) and $\mathscr{A}$ is the Borel $\sigma$-algebra on $\mathbb{R}$ (or a $\sigma$-algebra that contains the Borel sets).
Now, if we equip $C_c = \bigcup_{K \text{ compact}} C_c(K)$ with the inductive limit topology where $C_c(K)$ is the space of continuous functions $f \in C_c$ (i.e. $f$ has domain $\mathbb{R}$) with $\text{supp}(f) \subseteq K$ normed by $||f||_K := \sup_{x \in \mathbb{R}} |f(x)| = \sup_{x \in K} |f(x)|$, then $C_c$ is a locally convex space. The dual space is larger than in the case above and consists of all Radon "signed measures" (e.g. the Lebesgue measure $dx$ which is clearly not finite on $\mathbb{R}$ or $\sin(x) dx$ which is not a signed measure on a $\sigma$-algebra). I want to identify this dual space with an object coming from abstract measure theory. I think, the correct object is also a signed measure which is also countably additive BUT is defined on merely the $\delta$-ring generated by bounded intervals in $\mathbb{R}$ instead of a $\sigma$-algebra (the $\delta$-ring does not contain unbounded sets like the whole space).
Is this correct intuition? And if yes, does anyone know a reference for this fact?
(Of course one can generalize the ground space to be a more general topological space then $\mathbb{R}$, e.g. a locally compact second-countable Hausdorff space.)
EDIT: $C_c$ was primarily equipped with the wrong topology (the compact convergence). It is now correctly equipped with the inductive limit topology from its subspaces $C_c(K)$.
Best Answer
Edited following the edited question.
Yes, now the dual are all measures on the $\delta$-ring of bounded Borel sets. In your case they can be also considered as $\sigma$-finite measures on $\mathbb{R}$, since it is a countable union of compacts. In general, it's measures on the $\delta$-ring of Borel subsets of compact sets. In the topology of inductive limit, functionals are continuous when they are continuous on each $C_c(K)$, that is - measures on each $K$, from where it all follows.