Daniell-Loomis Integrals

Abstract

In [2] and [3] for arbitrary nonnegative linear functionals on functions vector lattices an integral extension of Lebesgue power has been discussed. Here we generalize this extension process, prove convergence theorems using a suitable “local convergence in measure,” discuss measurability and give characterizations by equality of upper and lower integrals. Riemann-μ, abstract Riemann-Loomis and Bourbaki integrals are subsumed. 0. Introduction. For a semi-ring Ω of sets from an arbitrary set X and μ : Ω → [0,∞[, only finitely additive, an analogue R1(μ,R) to the space L(μ,R) of Lebesgue-μ-integrable functions was introduced by Loomis [11]; this has been extended to Banach space-valued functions by Dunford-Schwartz [4], and in more general form in [6, 7]. Analogues to the Daniell extension process, but without or with weaker continuity assumptions on the elementary integral, have been treated by Aumann [1], Loomis [11] and Gould [5]. The Daniell-Bourbaki integral extension has been generalized with the integral I : B → R introduced in [2], starting with any nonnegative linear functional I on a vector lattice B of real-valued functions on X. If Ω is a δ-ring, μ σ-additive, I = ∫ .dμ on B = step functions over Ω, then R1, L and B coincide modulo null functions [3, 9]. In Sections 2 and 3 we generalize the extension I|B → I|B to I|B → J |L by “localization,” using an appropriate local convergence in measure, which is very useful to obtain convergence theorems in a form analogous to the classical ones (some of which are not true for B). In Section 4 we give various descriptions of the set L of integrable functions, in particular a Darboux-type characterization on L is proved. Always R1 ⊂ L (not true for B), in general B has infinite codimension in L, even modulo null functions. We recall that the abstract space of integrable functions L is constructed similar to the Daniell L and which coincides with L in the Received by the editors on April 21, 1996. AMS Mathematics Subject Classification. 28C05, 26A42. Copyright c ©1997 Rocky Mountain Mathematics Consortium