L-valued integration

TL;DR

This paper develops a new integration theory for functions valued in Dedekind complete unital f-algebras and explores the completeness of associated L^p spaces.

Contribution

It introduces an integration framework for L-valued functions and establishes completeness results for L^p spaces in this setting.

Findings

Established integration theory for L-valued functions

Proved completeness of L^p spaces with L-valued measures

Extended classical integration concepts to algebra-valued functions

Abstract

We develop integration theory for integrating functions taking values into a Dedekind complete unital $f$-algebra $\mathbb{L}$ with respect to $\mathbb{L}$-valued measures. We then discuss and prove completeness results of $\mathbb{L}$-valued $L^p$-spaces.