Integrals and Valuations

TL;DR

This paper establishes a deep connection between integrals on Riesz spaces and valuations on their spectrum by constructing a biinterpretable pair of geometric theories, revealing an elementary and structure-aware correspondence.

Contribution

It introduces two geometric theories that are biinterpretable, linking integrals on Riesz spaces with valuations on their spectrum in a novel, elementary way.

Findings

Constructs a homeomorphism between integrals and valuations

Develops two biinterpretable geometric theories

Shows the constructions are elementary and structure-dependent

Abstract

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.