We investigate an optimal control problem governed by differential equations with Radon measure as data in presence of state constraints. By using the method of graph completion for the measure, the original problem can be solved by studying an equivalent reparameterized control problem involving absolutely continuous trajectories but with time-dependent state constraints. Then it is showed that the epigraph of the reparameterized value function can be characterized by a Hamilton-Jacobi equation without any controllability assumptions.