Integration in a convex linear topological space

Abstract

This paper is concerned with a theory of integration for functions with values in a convex linear topological space. We consider an integral which is essentially an extension to this general space of the integral studied by Garrett Birkhoff [1] in a Banach space. By imposing different convex neighborhood topologies on a Banach space, we obtain as instances of our integral those defined by Birkhoff [1], Dunford [2], Gelfand [3], and Pettis [4]. Let f(s) be a function on an abstract set S to the real numbers, and let a(o-) be a nonnegative measure function on an additive family of subsets , of S. A necessary and sufficient condition for the Lebesgue integral to exist is that for each e >0 there exist a partition A0 of S into a denumerable set of sets (oi) such that for any two orderings of these sets, (ai) and (U-2),