Necessary conditions of optimality, in the form of a maximum principle, are derived for a class of optimal control problems, certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation. State trajectories are interpreted as robust solutions of the dynamic equations, a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps. The maximum principle which is derived improves on earlier optimality conditions for problems of this nature, by allowing nonsmooth data, measurable time dependence, and a possibly time-varying constraint set for the conventional controls.