ADVANCES IN GEOMETRIC INTEGRATION

Abstract

Over the last decade considerable progress has been made in developing extensions of the Lebesgue integral which provide the Gauss-Green theorem for noncontinuously differentiable vector fields and remain invariant under groups of transformations including diffeomorphisms. One particular extension, an averaging process defined by W.F. Pfeffer, accomplishes this in the setting of bounded sets of bounded variation—the most general class of sets for which the notions of “surface area” and “exterior normal” can be profitably defined. In this survey article we recover all of the notable geometric features behind Pfeffer’s extension in the less forbidding (vs. bounded sets of bounded variation) setting of figures, i.e. finite unions of compact intervals.