Generalized vectorial Lebesgue and Bochner integration theory

TL;DR

This paper develops a generalized theory of Lebesgue and Bochner integration using measures on prerings in abstract spaces, eliminating the need for topological structures and synthesizing classical results in measure theory.

Contribution

It introduces a measure-based framework on prerings for Lebesgue and Bochner spaces, broadening the scope of integration theory without topological constraints.

Findings

Measures on prerings generalize classical measures

Construction of Lebesgue-Bochner spaces without topology

Unified synthesis of classical measure theory results

Abstract

This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos, and other contributors to this theory. The construction of the theory is based on the notion of a measure on a prering of sets in any abstract space X. No topological structure of the space X is required for the development of the theory. Measures on prerings generalize the notion of abstract Lebesgue measures. These measures are readily available and it is not necessary to extend them beforehand onto a sigma-ring for the development of the theory. The basic tool in the development of the theory is the construction and characterization of Lebesgue-Bochner spaces of summable functions as in the paper of Bogdanowicz, "A Generalization of the Lebesgue-Bochner-Stieltjes Integral and a New Approach to the Theory of Integration", Proc. of Nat. Acad. Sci. USA, Vol. 53, No. 3, (1965), p. 492--498