The Daniell Integral: Integration without measure

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.