The Daniell Integral: Integration without measure

TL;DR

This paper explores Daniell's 1918 integral theory, which offers an alternative to measure-based integration by using linear functionals on vector lattices, with historical context and practical examples.

Contribution

It clarifies Daniell's integration approach, relating it to measure theory, and demonstrates its application to complex spaces through historical and practical examples.

Findings

Daniell's integral generalizes measure-based integration.

The theory applies to spaces beyond standard measure techniques.

Historical examples illustrate the practical utility of Daniell's approach.

Abstract

In his 1918 paper 'A General Form of Integral', Percy John Daniell developed a theory of integration capable of dealing with functions on arbitrary sets. Daniell's method differs from the measure-theoretic notion of integration. Linear functionals over vector lattices were considered as the fundamental objects on which he built the theory, rather than measures over sets. In this document, we explore Daniell's concept of integration and how his theory relates to the measure-theoretic notion of integration. We paint a picture of the historical context surrounding Daniell's ideas. Furthermore, we present examples due to Norbert Wiener, where the Daniell integral was employed on spaces too general for the standard integration techniques of the time.