A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology

This article begins by deriving a measure-theoretic decomposition of continuous linear functionals on $C(X)$, the space of all real-valued continuous functions on a metric space $(X,d)$, equipped with the topology ${\tau }_{\mathcal{B}}$ of uniform convergence on a bornology $\mathcal{B}$. We characterize the bornologies for which $(C(X),{\tau }_{\mathcal{B}}{)}^{\ast }=(C(X),{\tau }_{\mathcal{B}}^s{)}^{\ast }$, where ${\tau }_{\mathcal{B}}^s$ represents the topology of strong uniform convergence on $\mathcal{B}$. Furthermore, we examine the normability of ${\tau }_{ucb}$, the topology of uniform convergence on bounded subsets, on $(C(X),{\tau }_{\mathcal{B}}{)}^{\ast }$, and explore its relationship with the operator norm topology. Finally, we derive a topology on measures that shares a connection with $(C(X),{\tau }_{\mathcal{B}}{)}^{\ast }$ when endowed with ${\tau }_{ucb}$.