Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space $$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert f \rVert_\infty := \max_{x \in K} \lVert f(x) \rVert. $$ In the case, $X = \mathbb C$, it is well known that the dual space is given by the space of all regular Borel measures $\operatorname{rca}(K)$ on the space $K$.

Now I think one can not hope, in the general case that $X$ is a Banach space at least, for this characterization to carry over to the vector-valued case as one needs some kind of lattice structure to define regularity for vector valued measures. Moreover, if one looks at the $L^p$-situation, $1 \leq p <\infty$, one has $L^p(\Omega; X)' = L^{p'}(\Omega; X')$ if and only if the space $X$ has the Radon-Nikodym property. So a natural conjecture for me would be something like the following:

If $X$ is a Banach lattice, maybe with some additional Banach space or lattice properties, e.g., Radon-Nikodym or order continuity of the norm, then one has $C(K; X)' = \operatorname{rca}(K; X')$, where $\operatorname{rca}(K; X)$ denotes the space of $X'$-valued regular Borel measures.

I would expect that people asked and solved this question already. So my question is if a result of this kind of flavour is known? Moreover, are there good references for this kind of results? Thanks in advance!