pseudoparadox in measure theory

The interval [0,1) can be subdivided into an countably infiniteMathworldPlanetmath collectionMathworldPlanetmath of disjoint subsets Ai,i=0,1,2, such that, by translating each set one obtains a collection of disjoint sets Bi such that i=1Bi=[0,2)

This paradoxMathworldPlanetmath challenges the naive “” notion that if one a set into a countableMathworldPlanetmath number of pieces and reassembles them, the result will have the same measureMathworldPlanetmath as the original set.

The resolution to this and similar paradoxes lies in the fact that the sets Ai were not defined constructively. To show that they exist, one needs to appeal to the non-constructive axiom of choiceMathworldPlanetmath. What the paradox shows is that one can’t have one’s cake and eat it too — either one can cling to the naive “” picture and forego non-constructive techniques as the intuitionist school of mathematics does, or else if, like the majority of mathematicians, one wants to keep the powerful tools provided by non-constructive techniques in set theoryMathworldPlanetmath, one must give up the naive notion that every set is measurable and limit “” to operationsMathworldPlanetmath involving measurable setsMathworldPlanetmath.

It might be worth mentioning that it is essential that there be an infiniteMathworldPlanetmathPlanetmath number of sets Ai. As an elegant argument posted by jihema shows, it is not possible to find a finite collection of disjoint subsets of [0,1) such that a union of translations of these subsets equals [0,2). In higher dimensions, the situation is worse because, as Banach and Tarski showed, it is possible to derive analogous paradoxes involving only a finite number of subsets.

Title pseudoparadox in measure theory
Canonical name PseudoparadoxInMeasureTheory
Date of creation 2013-03-22 14:38:40
Last modified on 2013-03-22 14:38:40
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Theorem
Classification msc 28E99
Synonym one-dimensional Banach-Tarski paradox
Related topic VitalisTheorem
Related topic BanachTarskiParadox