A uniformly continuous linear extension principle in topological vector spaces with an application to Lebesgue integration

TL;DR

This paper introduces a new uniformly continuous linear extension principle in topological vector spaces, simplifying the construction of Lebesgue integrals for Banach space valued functions and deriving key theorems as straightforward consequences.

Contribution

It presents a novel extension principle that streamlines the construction of Lebesgue integrals and simplifies proofs of fundamental theorems in measure theory.

Findings

Simplified construction of Lebesgue integral for Banach space valued maps

Derivation of Vitali Convergence Theorem as a consequence

Derivation of Riesz-Fischer Theorem as a consequence

Abstract

We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali Convergence Theorem and the Riesz-Fischer Theorem are shown to follow as easy consequences from our construction.