- Hausdorff paradox
In
mathematics , the Hausdorff paradox, named afterFelix Hausdorff , states that if you remove a certain countable subset of thesphere "S"², the remainder can be divided into three subsets "A", "B" and "C" such that "A", "B", "C" and "B" ∪ "C" are all congruent. In particular, it follows that on "S"² there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal.The paradox was published in 1914. [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.] 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. The structure of the group of rotations on the sphere plays a crucial role here — this fact is not true on the plane or the line. In fact, it is possible to define "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 "area", however, is only finitely additive, so it is not at all a measure. In particular, it implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal
Lebesgue measure .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 on the real line which assigns a size to "all" bounded subsets of real numbers. This is very similar in nature to theVitali set .ee also
*
Vitali set
*Banach–Tarski paradox References
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