Th\'eorie de la mesure dans les lieux r\'eguliers. ou : Les intersections cach\'ees dans le paradoxe de Banach-Tarski

TL;DR

This paper explores a measure theory within the framework of locales, a generalization of topological spaces, showing that paradoxical decompositions become compatible with measurability, thus challenging classical notions of non-measurable sets.

Contribution

It introduces a measure theory on locales that reconciles the axiom of choice with universal measurability, avoiding paradoxical decompositions.

Findings

Measure extension to sub-locales is sigma additive outside regular measure.

Paradoxical partitions are replaced by hidden intersections in locales.

Non-measurable sets are eliminated in the locale framework.

Abstract

It is well known that axiom of choice implies the existence of non-measurable sets for Lebesgue's measure on R as well as the existence of "paradoxical" decompositions of the unit ball of R^3 (Banach-Tarski). This is generally interpreted as the price to be paid for the numerous services provided by this axiom. The theory proposed by Olivier Leroy shows that we can have simultaneously axiom of choice and " everything is measurable " it takes place within the framework of "locales" which are particular cases of Grothendieck's toposes : a "locale" is just a poset which has the formal properties of the poset of open subsets of a topological space. "Locales" have already been the object of numerous studies (cf for example "Sheaves in Geometry and Logic" of S.Mac Lane and I.Moerdijk. Springer 92.). One of the remarkable aspects of this theory is that it applies in a relevant way to the usual topological spaces in which it shows up " non standard sub-spaces " (sub-locales)); with for consequence that the intersection (in the meaning of locale) of ordinary sub-spaces is not anymore necessarily a (ordinary) sub-space. We have for example a sub-locale of R (called generic sub-locale of R) which is the intersection of dense open sets and which is still dense (although pointless). The most striking result is doubtless that the natural continuation of the measure of Lebesgue (on [0,1] for example) on the open sets to the totality of sub-locales of [0,1] is a sigma additive outside regular measure The "paradoxical " partitions which gave non-measurable subsets in the classical context are no more partitions in the context of locales. There are hidden intersections.... Jean-Malgoire et Christine Voisin