Abstract Integration in Net Convergence Structures
Alexandre Reggiolli Teixeira
TL;DR
This work develops a broad, unifying framework for integrating vector-valued functions with respect to arbitrary measures in spaces endowed with net convergence structures. It introduces the Abstract Net Riemann Integral, the $S^{*}$-integral, and Saks-type integrals, and shows Pettis integration can be represented via a net Riemann formulation; it proves a suite of convergence theorems (uniform, monotone, dominated) and Henstock-type results in both order and topological settings. It connects and compares many classical integrals (Lebesgue, Sion, Kolmogorov $S$, Henstock-Kurzweil, etc.) as special cases, and discusses regular integrators and subset properties, culminating in a proposed classification and open problems. The framework advances integration theory in spaces with general convergence (Riesz spaces, lattice-normed spaces) and offers tools for unifying disparate approaches across analysis, probability, and vector-valued measures.
Abstract
In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a convergence structure given by nets. This covers both the topological and order based convergences in the literature. We then show that this integral satisfies most of the common properties of the objects that comprises integration theory. By establishing a generalized notion of summability on Riesz spaces and an integral built upon countable partitions of the base space, we then stablish some uniform, monotone and dominated convergence theorems for the refereed integrals, as well as a non-topological or order based Henstock Lemma and a general convergence theorem based on the notion of conjugated lattice seminorms. An application of these theorems is made to prove various equivalences concerning the Lebesgue, for which we give a brief survey, Saks and Riemann type integrals in partially ordered and topological vector spaces presented in the literature, for which we also make a thorough review. We finish the article with a possible way of classifying general integration procedures defined in abstract convergence structures, and pose some open problems based on them.