A General Form of Integral

Introduction. The idea of an integral has been extended by Radon, Young, Riesz* and others so as to include integration with respect to a function of bounded variation. These theories are based on the fundamental properties of sets of points in a space of a finite number of dimensions. In this paper a theory is developed which is independent of the nature of the elements. They may be points in a space of a denumerable number of dimensions or curves in general or classes of events so far as the theory is concerned. It follows that, although many of the proofs given are mere translations into other language of methods already classical (particularly those due to Young), here and there, where previous proofs rested on the theory of sets of points, new methods have been devised (see, for example, theorems 3(3), 3(4), 5(1)). Mooret has developed a theory of a similarly general nature, but restricts himself to the use of relatively uniform sequences. This concept is not used nor is it necessary in the following paper. We consider a group of elements p, which may be whatever we choose, and certain classes of functions f(p) of those elements so that to every element p of the group there exists a real number f(p) (which may be infinite in certain cases). To each function of a certain class there corresponds a real number S(f ) or I(f ) which is defined so as to satisfy certain conditions. S(f ) is a generalized Stieltjes integral, while I(f ) may be called the positive integral and the latter possesses correlates of nearly all the properties of the Lebesgue integral. It is shown that any I-integral is an S-integral while any S-integral is expressible as the difference of two I-integrals. Two symbols have been taken over from symbolic logic, namely those for logical sum and logical product. The concepts involved are used extensively by Young and by the author and the symbols have been introduced to save space and to clarify the reasoning. The reader is referred also to Hildebrandtj for references to an exten-