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Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory.

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Dedicated to Marston Morse 1. Introduction and statement of the problem. The importance of the spectral reduction theory for bounded and unbounded selfadjoint and normal operators in Hubert space is amply demonstrated by its diverse applications to such far reaching fields of mathematics as the theories of topological groups, almost periodic functions, harmonic analysis, and selfadjoint boundary value problems. The problems centering around the reduction theory for nonnormal operators are among the most important problems in the theory of linear operators. Notable among the many early contributions to such problems were those of I. Fredholm [20] in 1903 and G. D. Birkhoff [5] in 1908. Fredholm discussed a certain class of linear integral equations and Birkhoff, a class of linear differential boundary value problems on a finite interval. The operators discussed in the Fredholm theory are compact and have spectra which are at worst convergent sequences. The corresponding spectral resolutions need not be countably additive, or, what amounts to the same thing, the eigenvalue expansions need not be unconditionally convergent. The Fredholm theory was later given a more abstract basis, stated in operator form and free of determinant theory, by F. Riesz [32], J. Schauder [33], and T. H. Hildebrandt [22]. The deep and comprehensive work of Birkhoff on eigenvalue expansions associated with (not necessarily selfadjoint) differential operators of arbitrary order strongly suggests that, except for certain irregular cases, linear differential boundary value problems on a finite interval will have unconditionally convergent (in Hilbert space) eigenvalue expansions. That this is indeed the case is shown by the work of J. T. Schwartz 1 and H. P. Kramer [24] who have given Birkhoff's results in an abstract linear operator form. The general formulation shows that the expansion theory is valid for operators whose analytical expressions may involve integral and difference operators as well as other types of terms. The recently announced results of M. A. Neumark [28; 29; 30] on singular differ-

In the last few years the work of Gelfand [17, 18], $ Kantorovitch [23, 24, 25], Dunford [9, 10], Vulich [24, 25,42] and others has shown that in developing a representation theory for various classes of linear operations! among Banach spaces [1] effective use can be made of abstract functions and integrals, just as the general linear functional over certain 7i-spaces were earlier discovered to be representable in terms of numerical functions and the integrals of numerical functions. This is especially true for operations defined to a general 73-space X from a Lebesgue space, that is, from a space consisting of a class of Lebesgue-integrable numerical functions. To obtain representations for operations of this sort it was found that ready application could be made of various integrals of the Lebesgue type that have been defined for functions taking their values in X.In the present paper we wish to communicate a representation theory for several types of operators mapping a space L(S), consisting of the real functions that are Lebesgue-integrable over an abstract aggregate 5 with respect to a fixed class of subsets of 6" and a fixed measure function [29,34], into an arbitrary 73-space X. The representations will be given in terms of abstract integrals and kernel integrals. The general approach is not new, for it is based on the methods introduced by Gelfand [18] and Dunford [9] to obtain such theorems when 5 is a bounded real interval. However, in order to extend these known results to the case of an arbitrary 5 new devices are required since the earlier results were proved by Euclidean methods. In most instances we have been able to make the extension; this has been accomplished by generalizing the Radon-Nikodym theorem [29,34] X Numbers in brackets refer to the references at the end. Gelfand's paper [18] is the thesis which he presented in June, 1935. § Hereafter for linear operations we shall use the briefer terms operations or operators since these are the only operations that will come into consideration. An operation, or operator, is thus understood to be a distributive continuous mapping of one 5-space into another. 323License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NELSON DUNFORD AND B. J. PETTIS[May ous* is also considered. By means of these representation theorems new information is given concerning certain types of operators. This information in turn yields a uniform mean ergodic theorem for weakly c.c.f operations in L(S) and an application to Markoff processes. In addition it provides results which may be of interest in the theory of integral equations. In terms of both abstract integrals and kernel integrals a fairly complete representation theory is given for operations mapping L(S) into the Lebesgue classes L"(T), 1 g q g 00, where T is another aggregate; the types considered are the general, the separable,!: the weakly c.c, and the c.c. operations. Those results dealing with arbitrary S and T will have as immediate corollaries the c...