Dual space of the space of finite measures

In the case of measures on a compact space, you are talking about the bidual of $C(K)$. This space was investigated in detail by S. Kaplan who wrote a series of long papers on it in the Transactions---easily available online. He also produced a book summarising his results. The natural extension for completely regular spaces would be the bidual of the space of bounded, continuous functions thereon, with the strict topology. This is certainly an interesting space and many of Kaplan's results carry over in suitably modified form but nobody has written this up to my knowledge.

Well, your space of measures is isometric to $L^1(\mu)$ for some (probably very big, non-sigma-finite) measure $\mu$. So it is enough to know what is the dual of an $L^1$ space.