Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded *uniformly* continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?

I would guess it can be described as some space of measures. I would even be interested in the case $X=\mathbb{R}$.

Obviously if $X$ is compact this is just the signed (or complex) Radon measures on the Borel $\sigma$-algebra of $X$. If $d$ is a discrete metric then we have all finitely additive measures on $X$. But more generally I do not know.

**Edit:** If $C_b(X)$ is the Banach space of all bounded continuous functions on $X$, we of course have $C_u(X) \subset C_b(X)$ as a closed subspace, and we know that $C_b(X)^*$ can be identified with the space of finite, regular, finitely additive signed Borel measures on $X$. Certainly each such measure gives us a continuous linear functional on $C_u(X)$, so we have a map $C_b(X)^* \to C_u(X)^*$ which is just the restriction map, but it is not injective. Conversely, by Hahn-Banach each bounded linear functional on $C_u(X)$ extends to a bounded linear functional on $C_b(X)$, but not in a canonical way.

Also, it is clear that in general $C_u(X)^*$ contains more than just the countably additive measures, since e.g. if $X=\mathbb{R}$ it contains some Banach limits. So we have all the countably additive finite Radon measures, but not all the finitely additive finite regular measures. This is why I would imagine that $C_u(X)^*$ consists of all finitely additive measures satisfying some condition that is more than "regular" but less than "countably additive". But I have no idea what it might be.

As mentioned in comments, I am happy to know about nontrivial special cases: $X$ locally compact, $X$ complete and separable, etc.