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Fubini's Theorem for Daniell Integrals

Götz Kersting, Gerhard Rompf

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.

Fubini's Theorem 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.
Paper Structure (3 sections, 4 theorems, 19 equations)

This paper contains 3 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction and main results
  2. Proofs
  3. Remark.

Key Result

Theorem 1

For any two integrals $J:G\to \mathbb R$ and $K:H\to \mathbb R$ there is a unique integral $I:L(G^1\otimes H^1) \to \mathbb R$ fulfilling for all $g\in G^1$, $h\in H^1$ the equation Moreover, any $f\in L(G^1\otimes H^1)$ meets the conditions (a) and (b), and we have

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • Proposition 2
  • proof : Proof of Theorem \ref{['thm2']}