Fubini's Theorem for Daniell Integrals

TL;DR

This paper extends Fubini's theorem to Daniell integrals, demonstrating that iterated integrals can always be formed and interchanged, thus generalizing product integrals within Daniell's framework.

Contribution

It establishes the full generality of Fubini's theorem for Daniell integrals using advanced density theorems and tensor product techniques.

Findings

Iterated integrals can always be formed in Daniell integration.

Order of integration can be interchanged in Daniell integrals.

Fubini's theorem holds in full generality for Daniell integrals.

Abstract

We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini-Stone Theorem.