Oriented Measures and the Bang-bang Principle

We introduce the notion of an oriented measure. The interior of the range R of a non-atomic oriented measure mu is {mu(E) : chi(E) has n discontinuity points} and moreover mu(E) is-an-element-of partial-derivative R double-line arrow pointing left and right chi(E) has less than n - 1 discontinuity points. Given a solution x to x(n) + a(n-1) (t) x(n-1) +... + a1 (t) x' + a0 (t) x is-an-element-of [phi1, phi2] on [a, b] there exist two bang-bang solutions y and z having a contact of order n with x at a and b such that y less-than-or-equal-to x less-than-or-equal-to z. An application to the calculus of variations yields a density result.