Theory of a Class of Locally Convex Vector Lattices Which Include the Lebesgue Spaces

Abstract

In this paper is presented the theory of a class of locally convex lattices (L-lattices) of real functions which generalize the classical Lebesgue spaces. The monotone and dominated convergence theorems for convergence almost everywhere and sequential and order completeness of such lattices are established. These results are obtained through characterization of linear lattices of functions closed under pointwise or dominated convergence everywhere and closed under Stone's operation f → f[unk]1. Such lattices are characterized in terms of measurability with respect to sigma or delta rings.

Application to the theory of Lebesgue integrals is given, permitting one to obtain the classical theory of the integral and of the L_p spaces directly from the axioms of the integral. Representations of L-lattices in terms of Lebesgue integrals and extensions of L-lattices through addition of null functions are given.