Summary: For a general optimal control problem with a state constraint, we propose a proof of the maximum principle based on a \(v\)-change of the time variable \(t \mapsto \tau \), under which the original time becomes yet another state variable subject to the equation \(dt /d \tau = v( \tau )\), while the additional control \(v ( \tau ) \geq 0\) is piecewise constant and its values are arguments of the new problem. Since the state constraint generates a continuum of inequality constraints in this problem, the necessary optimality conditions involve a measure. Rewriting these conditions in terms of the original problem, we get a nonempty compact set of collections of Lagrange multipliers that fulfil the maximum principle on a finite set of values of the control and time variables corresponding to the \(v\)-change. The compact sets generated by all possible piecewise constant \(v\)-changes are partially ordered with respect to inclusion, thus forming a centered family. Taking any element of their intersection, we obtain a universal optimality condition, in which the maximum principle holds for all values of the control and time.