Measure theory via Locales

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 $\sigma$-algebras. As an application we obtain a functorial construction of the induced measure $\mu_$ on the locale of sublocales $\mathfrak{Sl}(X)$ of a Hausdorff space $X$ equipped with a Radon measure $\mu$, which in particular shows that $\mu_$ 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.