INTEGRATION IN NONCOMMUTATIVE SYSTEMSi1)

In this paper we introduce an abstract integral which includes both product integrals and additive integrals. Thus our theory of Riemann integration embraces both the product integrals of Volterra [l, 2, 3](2), Birkhoff [3], and Masani [l] and the classical additive integrals of Riemann [l] and Stieltjes [l]. Similarly our theory of Lebesgue integration includes the Lebesgue product integrals of Schlesinger [l] and Birkhoff [3] as well as the familiar Lebesgue integral [Lebesgue l] and the additive integrals of Birkhoff [l] and Price [l]. This unified treatment is made possible by a suitable choice of the objects to be integrated—the "differentials" of Definition 3.15. We define integrals in a system with but a single operation. The use of only one operation simplifies appreciably the theory of product integration. Certain new results on product integration appear as corollaries of our general theory. Thus, for functions with values in a normed ring, we prove the existence of the Stieltjes product integral of a continuous function with respect to a function of bounded variation. New theorems on the Lebesgue product integrability of Riemann product integrable functions are obtained. The existence theorem for Birkhoff's Riemann product integral in a nondistributive system is extended. The paper is divided into four parts. In Part I, after some notational preliminaries, we introduce our concept of a differential and define certain properties of differentials. The other novelties in this part are two concepts that serve to replace convexity in studying integration in systems where convexity has no meaning. These are "stability of multiplication" (Definitions 3.21 and 3.22) and the use of subpartitions in defining integral ranges (Definitions 3.16-3.19(3). Parts II and III are respectively studies of Riemann and Lebesgue integration in our system. The principal results are generalizations of (i) the theorem that an almost everywhere continuous function is Riemann integrable and (ii) the theorem that a Riemann integrable function is Lebesgue integrable.