Measuring definable sets in o-minimal fields

2014/02/19 by Jana Maříková, Maříková, Jana, Masahiro Shiota +1
Mathematics · #03C64 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Mathematical Dynamics and Fractals #math.LO #msc:03C64

paper · pdf · doi:10.48550/arxiv.1402.4787

openalex publication_date 2014/02/19 · arxiv created 2014/04/28 · arxiv updated 2014/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field R. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of R. We show that every measurable subset of Rⁿ with non-empty interior has positive measure, and that the measure is preserved by definable C¹-diffeomorphisms with Jacobian determinant equal to ± 1.

Related

• The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field
• A real-valued measure on non-Archimedean field extensions of ℝ
• Expansions of the real field by open sets: definability versus interpretability
• Definable completeness of P-minimal fields and applications
• O-minimal fields with standard part map