[Un sous-corps des réels définissant un ensemble non mesurable]
In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield $K$ of the real numbers that lacks several such tameness properties. As our main result, we present a first-order formula in the language of rings defining a set in $K$ that is not Lebesgue measurable. Moreover, $K$ has the independence property and admits both archimedean and non-archimedean orderings.
Dans le domaine de la géométrie modérée, différentes conditions de modération de théorie des modèles sont établies et leurs relations sont analysées. Nous construisons un sous-corps $K$ des nombres réels qui ne satisfait pas plusieurs de ces propriétés de modération. Comme résultat principal, nous présentons une formule du premier ordre dans le langage des anneaux définissant un ensemble dans $K$ qui n’est pas Lebesgue-mesurable. De plus, $K$ a la propriété de l’indépendance et admet à la fois des ordres archimédiens et non archimédiens.
Keywords: Borel measurable, first-order definable, Lebesgue measurable, NIP, non-measurable, tame geometry
Lothar Sebastian Krapp 1 , 2 ; Matthieu Vermeil 3 , 4 ; Laura Wirth 2
CC-BY-ND 4.0
Lothar Sebastian Krapp; Matthieu Vermeil; Laura Wirth. A Subfield of the Reals Defining a Non-Measurable Set. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 71-85. doi: 10.5802/pmb.71
@article{PMB_2026____71_0,
author = {Lothar Sebastian Krapp and Matthieu Vermeil and Laura Wirth},
title = {A {Subfield} of the {Reals} {Defining} a {Non-Measurable} {Set}},
journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
pages = {71--85},
year = {2026},
publisher = {Presses universitaires de Franche-Comt\'e},
doi = {10.5802/pmb.71},
language = {en},
url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.71/}
}
TY - JOUR AU - Lothar Sebastian Krapp AU - Matthieu Vermeil AU - Laura Wirth TI - A Subfield of the Reals Defining a Non-Measurable Set JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2026 SP - 71 EP - 85 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.71/ DO - 10.5802/pmb.71 LA - en ID - PMB_2026____71_0 ER -
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