Measure theory via Locales
Georg Lehner
TL;DR
The paper develops a comprehensive locale-theoretic foundation for measure theory by replacing $σ$-algebras with valuations on frames and leveraging Grothendieck topologies to present measure algebras from Radon measures. It constructs a functorial induced measure $\mu_*$ on the locale of sublocales $\mathfrak{Sl}(X)$ for a Hausdorff space $X$ with a Radon measure $\mu$, proving invariance under measure-preserving homeomorphisms, and introduces measurable locales in duality with commutative von Neumann algebras via $L^\infty$. A representation theorem shows every measurable locale arises from a regular content on a locally compact Hausdorff space, enabling a geometric interpretation through the Lebesgue locale for manifolds. The framework provides a robust, point-free perspective on measure, integration, and conditional structure, with strong functorial properties and connections to classical measure theory through inner topologies and sublocale constructions.
Abstract
We present an approach to measure theory using the theory of locales. This includes concrete constructions of measure algebras associated to Radon measures, such as the Lebesgue measure on $\mathbb{R}^n$, via Grothendieck topologies constructed from valuations, that circumvent the classical approach via $σ$-algebras. As an application we obtain a functorial construction of the induced measure $μ_*$ on the locale of sublocales $\mathfrak{Sl}(X)$ of a Hausdorff space $X$ equipped with a Radon measure $μ$, which in particular shows that $μ_*$ is invariant under measure-preserving homeomorphisms. We furthermore give a construction of the measurable locale associated to a smooth manifold, functorial in submersions, as well as comparison results to classical measure theory.