The Daniell Integral

TL;DR

This paper presents a straightforward approach to the Daniell integral, focusing on direct approximation of integrable functions without auxiliary spaces, simplifying the understanding of its fundamental properties.

Contribution

It introduces a simple, direct method for analyzing the Daniell integral, avoiding the traditional auxiliary space framework.

Findings

Basic properties of the Daniell integral are established.

The approach simplifies the understanding of the integral's fundamental aspects.

Approximation of functions by absolutely convergent series is effective.

Abstract

The basic properties of the Daniell integral are presented. We do not use the standard approach of introducing auxiliary spaces of the "over-functions" and "under-functions." Instead, we use a simple and direct approach based on approximating integrable functions by absolutely convergent series of simple functions.