Date links per wp:mosnum/Other |
Eric Kvaalen (talk | contribs) Link to Banach measure |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
In [[mathematics]], the '''Hausdorff paradox''', named after [[Felix Hausdorff]], states that if you remove a certain [[ |
In [[mathematics]], the '''Hausdorff paradox''', named after [[Felix Hausdorff]], states that if you remove a certain [[Countable set|countable]] subset of the [[sphere]] ''S''², the remainder can be divided into three disjoint subsets ''A'', ''B'' and ''C'' such that ''A'', ''B'', ''C'' and ''B'' ∪ ''C'' are all [[Congruence (geometry)|congruent]]. In particular, it follows that on ''S''² there is no [[Measure_(mathematics)#Generalizations|finitely additive measure]] defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of ''A'' is both 1/3 and 1/2 of the non-zero measure of the whole sphere). |
||
The paradox was published in 1914 |
The paradox was published in [[Mathematische Annalen]] in 1914<ref> |
||
F. Hausdorff, ''[http://docserver.digizeitschriften.de/contentserver/contentserver?command=docconvert&docid=362514 Bemerkung über den Inhalt von Punktmengen]'', [http://gdz-srv3.sub.uni-goettingen.de/cache/toc/D25917.html Mathematische Annalen], vol 75. (1914) pp. 428-434. |
|||
</ref> and also in Hausdorff's book, [[Grundzüge der Mengenlehre]], the same year. The proof of the much more famous [[Banach–Tarski paradox]] uses Hausdorff's ideas. |
|||
| ⚫ | |||
This paradox shows that there is no finitely additive measure on a sphere defined on ''all'' subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no ''countably'' additive measure defined on all subsets.) The structure of the [[SO(3)|group of rotations on the sphere]] plays a crucial role here — the statement is not true on the plane or the line. In fact, as was later shown by [[Banach]],<ref> |
|||
Sometimes the '''Hausdorff paradox''' refers to another theorem of Hausdorff which was proved in the same paper. This theorem states that it is possible to "chop up" the [[unit interval]] into countably many pieces which (by translations only) can be reassembled into the interval of length 2. Hausdorff described these constructions in order to show that there can be no non-trivial, translation-invariant [[Measure (mathematics)|measure]] on the real line which assigns a size to ''all'' bounded subsets of real numbers. This is very similar in nature to the [[Vitali set]]. |
|||
[[Stefan Banach]], [http://matwbn.icm.edu.pl/ksiazki/fm/fm4/fm412.pdf "Sur le problème de la mesure"], [[Fundamenta Mathematica]] 4, pp. 7-33, [[1923]]; Banach, [http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf "Sur la décomposition des ensembles de points en parties respectivement congruentes"], Theorem 16, Fundamenta Mathematica 6, pp. 244-277, [[1924]]. |
|||
| ⚫ | </ref> it is possible to define an "area" for ''all'' bounded subsets in the Euclidean plane (as well as "length" on the real line) such that congruent sets will have equal "area". (This [[Banach measure]], however, is only finitely additive, so it is not a [[Measure (mathematics)|measure]] in the full sense, but it equals the [[Lebesgue measure]] on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are [[Banach–Tarski paradox|equi-decomposable]] then they have equal area. |
||
==See also== |
==See also== |
||
*[[Vitali set]] |
|||
*[[Banach–Tarski paradox]] |
*[[Banach–Tarski paradox]] |
||
Revision as of 14:41, 9 November 2008
In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S², the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent. In particular, it follows that on S² there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of A is both 1/3 and 1/2 of the non-zero measure of the whole sphere).
The paradox was published in Mathematische Annalen in 1914[1] and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas.
This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no countably additive measure defined on all subsets.) The structure of the group of rotations on the sphere plays a crucial role here — the statement is not true on the plane or the line. In fact, as was later shown by Banach,[2] it is possible to define an "area" for all bounded subsets in the Euclidean plane (as well as "length" on the real line) such that congruent sets will have equal "area". (This Banach measure, however, is only finitely additive, so it is not a measure in the full sense, but it equals the Lebesgue measure on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal area.
See also
References
- ^ F. Hausdorff, Bemerkung über den Inhalt von Punktmengen, Mathematische Annalen, vol 75. (1914) pp. 428-434.
- ^ Stefan Banach, "Sur le problème de la mesure", Fundamenta Mathematica 4, pp. 7-33, 1923; Banach, "Sur la décomposition des ensembles de points en parties respectivement congruentes", Theorem 16, Fundamenta Mathematica 6, pp. 244-277, 1924.