On integration in vector spaces

Abstract

Introduction.Several authors ([3]-[10], inclusive; [I5])f have already given generalized Lebesgue integrals for functions x(s) whose values lie in a Banach space (P-space) 3c.fIn the following pages another definition, § based on the linear functionals over 3c and on the ordinary Lebesgue integral, || will be given and its properties and relationships to other integrals discussed.We consider functions x(s) defined to a P-space X = [x] from an abstract space S= [s] possessing both a c-field 2 of "measurable" sets having S as an element and a non-negative bounded c.a. (completely additive) "measure" function a(E) defined over 2. Notational conventions are as follows: 3c denotes the P-space conjugate to 3c ([l], p. 188),/=/(x), g = g(x), • ■ ■ denote elements of 3c, and F = F(f), G = G(f), ■ ■ ■ elements of 3c, the conjugate space of 3c; and for real-valued functions Greek letters will be used.When the abstract functions x(s), y(s), • • ■ , or the real functions 4>(s), \p(s), ■ ■ ■ are considered as elements of a functional space, they will sometimes be written x(:), y(:) or <f>(:), \f/(:).The "zero" element of 3c will be denoted by d.The contents of the paper may be outlined as follows.The first section compares measurability of functions under the strong and weak topologies of 3c.The second defines the (3c) integral and utilizes an essential lemma, first stated by Orlicz, to prove the complete additivity and absolute continuity of the integral.The linear operations from Lp to 3c and from Ï to Lp defined by integrable functions are investigated in §3, and these results are expressed in terms of real-valued kernels in §7.Approximation and convergence theorems occupy §4 and lead to §5 and the relationships between the (3c) integral and the integrals given by other definitions.A few remarks on completely continuous transformations and on differentiation account for § §6 and 8, respectively, and four examples form §9. In conclusion a few open questions are cited.