Burkill-Cesari integrals of quasi additive interval functions

Abstract

The Burkill-Cesari B-C integrals arising in L. Cesari's theory of quasi additive vector-valued set functions need not be additive as functions of sets. It is shown in the present paper that these integrals satisfy quasi additivity and overadditivity properties. These properties are used to prove Banach-type differentiation theorems for B-C integrals defined on Euclidean spaces. Variants of Cesari's basic quasi additivity hypothesis and some simplifications in the formulation of the general theory are also discussed.