# pseudoparadox in measure theory

The interval $[0,1)$ can be subdivided into an countably infinite  collection  of disjoint subsets $A_{i},i=0,1,2,\ldots$ such that, by translating each set one obtains a collection of disjoint sets $B_{i}$ such that $\bigcup_{i=1}^{\infty}B_{i}=[0,2)$

The resolution to this and similar paradoxes lies in the fact that the sets $A_{i}$ were not defined constructively. To show that they exist, one needs to appeal to the non-constructive axiom of choice  . 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 theory  , one must give up the naive notion that every set is measurable and limit “” to operations  involving measurable sets  .

It might be worth mentioning that it is essential that there be an infinite   number of sets $A_{i}$. 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 PseudoparadoxInMeasureTheory 2013-03-22 14:38:40 2013-03-22 14:38:40 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Theorem msc 28E99 one-dimensional Banach-Tarski paradox VitalisTheorem BanachTarskiParadox