# pseudoparadox in measure theory

The interval $[0,1)$ can be subdivided into an countably infinite^{} collection^{} of disjoint subsets ${A}_{i},i=0,1,2,\mathrm{\dots}$ such that, by translating each set one obtains a collection of disjoint sets ${B}_{i}$ such that ${\bigcup}_{i=1}^{\mathrm{\infty}}{B}_{i}=[0,2)$

This paradox^{} challenges the naive “” notion that if one a set into a countable^{} number of pieces and reassembles them, the result will have the same measure^{} as the original set.

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 |
---|---|

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 |