On everywhere-defined integrals
Lester E. Dubins
Transactions of the American Mathematical Society
Hardly any finite integrals can be defined for all real-valued functions. In contrast, if infinity is admitted as a possible value for the integral, then every finite integral can be extended to all real-valued functions. Introduction. An S-integral is an order-preserving, linear functional defined for an ordered linear space, S. A natural problem would seem to be: for a given S, find all the S-integrals. Of course, in the special case in which S is the set of continuous, real-valued, functions
... on a compact space, then a well-known theorem of F. Riesz, and its generalizations, provides some answer. In this paper, I study another special case, namely that in which S is the space of all finite, real-valued functions defined on a set, and conclude that there are very few such integrals (Lemma 1, Theorem 1, and Corollary 1). Since it is convenient to have an integral defined for all functions, the question then arises as to the possibility of extending an integral to the space L of all functions if infinity is admitted as a possible value. This question has an affirmative answer, as is the purpose of §2 to demonstrate. 1. Throughout this paper, ñ is a fixed, nonempty set, % is a o*-field of its subsets, L is the vector lattice of all finite, real-valued, % -measurable functions defined on ß, and 5 is a linear subspace of L. Of course, L is ordered by the cone of all nonnegative elements of L. If, for some w G Q, P(f) = f(u>) for all / G L, then P is an evaluation. If Px,... ,Pn are evaluations, and /"..., tn are nonnegative, real numbers, then "2"xt¡P¡ is an elementary integral. As is the purpose of this section to show, for interesting %, there are no L-integrals other than elementary integrals. Call ai/£% trivial for P if P(U) = 0, and very trivial for P if, for every f EL which vanishes outside U, Pf = 0. Let Uc he the complement of U. Call reí reducible if 3U E %, U c T, such that neither U nor TUC is very trivial. If T is neither reducible nor very trivial, then T is an atom for P. If, for an atom T, Pf'= P(fT) for all / G L, then P is atomic. (The Received by the editors February 9, 1976. AMS (MOS) subject classifications (1970). Primary 28A30.