Hausdorff paradox

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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 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.

The paradox was published in 1914.^[1] 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 the Vitali set.