The representation of abstract integrals

Abstract

(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 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 4 from one function space Fo to another, Eo, can be obtained by coordinatewise integration in a product space. That is, there exist spaces X, Y such that Fo is isomorphic, in a sense to be defined below (?1) to a certain space of functions on XX Y, and F' is isomorphic to a space of functions on X, and under these isomorphisms 4 corresponds to the mapping f->f' where f'(x) =ff(x, y)dy, the integral being formed with respect to an ordinary (countably additive) c-finite numerical measure on Y. (The exact theorem is stated in 5.3 below.) The formal properties of the mapping 0, and the representation theorem just mentioned, entitle 0 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 F' consists of the functions defined on a single point.