A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem

Abstract

AbstractIn this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.