If a set-function $\mu'$ with certain properties (not important here) is defined on the set $K(X)$ of compact subsets of a topological space $X$, a regular measure $\mu$ can be constructed such that $\mu(K) = \mu'(K) \ \forall K \in K(X)$.
My question is: is it possible to construct a positive bounded regular measure $\mu$ from a positive bounded set-function $\mu'$ defined on the set $O(X)$ of open subsets of a topological space $X$ such that $\mu(O) = \mu'(O) \ \forall O \in O(X)$? If so, what properties should $\mu'$ have?
I believe that the definition should be $\mu (S) = \inf _{S \subseteq O} \mu'(O)$, the infimum being taken over all open subsets $O$ containing $S$, but I am interested in the possibility (and general idea) of the construction.