Integration in general analysis

defined and developed the theory of the Riemann integral for functions whose values are in a complete linear vector space.T. H. Hildebrandtf and S. Bochner have defined the Lebesgue integral for the same type of function.The present paper, which approaches the theory of the integral in a manner analogous to the Cantor definition of a real number, is concerned chiefly with the convergence of a sequence of integrals and is not as extensive in scope as that of Bochner which contains certain results pertaining to multiple integrals, Fourier series, and the class Lp.In what follows no use is made of the theory of integration for numerically valued functions other than a knowledge of the properties of an additive class of point sets and of a completely additive function on such a class.In fact the method when applied to such functions seems in many ways more direct than the classical one.The proofs in the section on types of convergence are omitted since they may be carried through exactly as in the case of real-valued functions.In the last section it is shown how the theory holds, with slight modifications, for a function having an arbitrary metric space as its domain and a Banach space for its range.1. Basis.A class of point sets is said to be additive if for every pair of sets E, D and every sequence {En} of disjoint sets in the class the sets E-D, X)2" are also in the class.A function a(E) on an additive class of sets A is said to be completely additive if for every sequence {"} of disjoint sets in A, <x(Y^Ei) =Y^aiEi).In what follows, A will be used to denote an additive class of point sets which contains all Borel measurable sets belonging to a fundamental bounded and closed interval / of a euclidean space of n dimensions.The notation a(E) will always be used for a completely additive function on A to the real number system and iE) will stand for the total variation of a on E. Radon|| has constructed such systems iA, a, ) corresponding to a * Presented to the Society,