I am having trouble with a certain technical part of solving a problem. The question can be formulated as the following:
Let $X$ be a locally compact Hausdorff space and $M(X)$ be the space of all regular Borel complex measures on $X$. Pick any $\mu, \nu \in M(X)$. We have that
$$\int f \ d\mu = \int f \ d\nu$$ for every bounded continuous function $f$ on $X$.
Is it sufficient to conclude that $\mu = \nu$? If not,then what additional condition is needed?
I believe that since $C_0(X) \subset C_b(X)$ and $C_0(X)$ is a separating family of functions, it should be true.