The representation of abstract integrals

Abstract

Introduction.This paper is primarily concerned with linear order-preserving mappings from one function-space to another.For example, let / be a (suitably restricted) function of two real variables, and consider the function f'(x)-ff(x, y)dy.The mapping f-f' is a linear, countably additive mapping from the space of (suitably restricted) functions of two variables to the space of functions of one variable; and it is "order-preserving" in the sense that positive functions are mapped into positive functions.The main result of this paper implies that this example is (roughly speaking) typical; under mild hypotheses, mostly of countability, every linear, countably additive, order-preserving mapping <f> from one function space Ao to another, F0', can be obtained by coordinatewise integration in a product space.That is, there exist spaces X, F such that F0 is "isomorphic," in a sense to be defined below ( §1) to a certain space of functions on ZX Y, and F¿ is isomorphic to a space of functions on X, and under these isomorphisms <j> corresponds to the mapping f-*f where f'(x) =Jf(x, y)dy, the integral being formed with respect to an ordinary (countably additive) <r-finite numerical measure on Y. (The exact theorem is stated in 5.3 below.)The formal properties of the mapping cf>, and the representation theorem just mentioned, entitle <j> to be called an "abstract integral."The ordinary Lebesgue integral (over a fixed set with a a-finite measure) is included as a special case, in which F0' consists of the functions defined on a single point.Abstract integrals have been considered from much this point of view before(1), and representation theorems for linear mappings between two function spaces are known in many special cases(2).But the emphasis in the present investigation is on rather different aspects of the problem, and it is believed that there is little overlap with previous work.Briefly, the present treatment is aimed at saying as much as possible about the structure of a single "abstract integral" <p, where <j> is to have the salient features of the Lebesgue definite integral, but with values which (instead of being numbers) lie in as general a function space as can be handled.The only essential restriction we impose on our function spaces is that they satisfy the countable chain condition.Such spaces have no useful norms, and no norm conditions will be Presented to the Society, September 5, 1953; received by the editors August 19, 1952.(') See [2, Chapter III], and references given there.(2) See [3], and references there given.