Sublinear functions of measures and variational integrals

The original purpose of this paper was to establish that certain non-parametric variational integrals may be considered as measures on the domain of definition of the admissible functions. This was accomplished by exhibiting an explicit formula for the functional involved, the formula containing within itself, in the case of the surface area integral, a proof both of Tonelli’s celebrated theorem on Lebesgue area and of a less well known but deeper result of Verchenko. It soon became apparent, however, that the basic techniques were considerably more general, and could, in fact, be phrased entirely apart from variational calculus simply as a method of generating measures in terms of other measures. We shall follow this more abstract course in the paper, and only turn directly to variational calculus in the final section. Consider a countably additive set function on a g-ring R of subsets of a basic set S. The function g will be supposed to have its values in a real Banach space , which for particular applications may be the real numbers R1, or the Euclidean number space R. Let (p) be a bounded sublinear functional on into the reals. By this we mean that for all p, q in