Measure theory via Locales

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.