Ulrik Buchholtz

arXiv.org

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 $σ$-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 meas