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bounded subsets in the
Euclidean plane (as well as "length" on the real line) in such a way that congruent sets will have equal "area". (This
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plays a crucial role here – the statement is not true on the plane or the line. In fact, as was later shown by
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subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no
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defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of
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on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are
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This paradox shows that there is no finitely additive measure on a sphere defined on
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492:"Sur la décomposition des ensembles de points en parties respectivement congruentes"
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29:
420:
494:, Theorem 16, Fundamenta Mathematicae 6: pp. 244–277, 1924.
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uses
Hausdorff's ideas. The proof of this paradox relies on the
126:, then the remainder can be divided into three disjoint subsets
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additive measure defined on all subsets.) The structure of the
490:, Fundamenta Mathematicae 4: pp. 7–33, 1923; Banach,
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400:, the same year. The proof of the much more famous
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385:of the non-zero measure of the whole sphere).
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509:"Bemerkung ĂĽber den Inhalt von Punktmengen"
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60:(the surface of a 3-dimensional ball in
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394:in 1914 and also in Hausdorff's book,
439:in the full sense, but it equals the
7:
242:. In particular, it follows that on
85:{\displaystyle {\mathbb {R} ^{3}}}
14:
421:group of rotations on the sphere
93:). It states that if a certain
488:"Sur le problème de la mesure"
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537:(Original article; in German)
388:The paradox was published in
447:then they have equal area.
601:
544:GrundzĂĽge der Mengenlehre
541:Hausdorff, Felix (1914).
507:Hausdorff, Felix (1914).
462: – Geometric theorem
397:GrundzĂĽge der Mengenlehre
294:{\displaystyle {B\cup C}}
271:finitely additive measure
231:{\displaystyle {B\cup C}}
466:Paradoxes of set theory
203:{\displaystyle {A,B,C}}
119:{\displaystyle {S^{2}}}
97:subset is removed from
53:{\displaystyle {S^{2}}}
575:Mathematical paradoxes
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513:Mathematische Annalen
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402:Banach–Tarski paradox
391:Mathematische Annalen
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262:{\displaystyle S^{2}}
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147:{\displaystyle {A,B}}
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580:Theorems in analysis
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378:{\displaystyle 2/3}
350:{\displaystyle 1/2}
322:{\displaystyle 1/3}
169:{\displaystyle {C}}
525:10.1007/bf01563735
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301:is simultaneously
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28:. It involves the
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18:Hausdorff paradox
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20:is a paradox in
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501:Further reading
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519:(3): 428–434.
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561:on ProofWiki
547:(in German).
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269:there is no
24:named after
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22:mathematics
569:Categories
472:References
176:such that
533:123243365
417:countably
285:∪
240:congruent
222:∪
95:countable
454:See also
238:are all
437:measure
531:
425:Banach
357:, and
30:sphere
529:S2CID
210:and
154:and
16:The
521:doi
429:all
413:all
408:.
571::
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517:75
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486:,
329:,
535:.
523::
373:3
369:/
365:2
345:2
341:/
337:1
317:3
313:/
309:1
288:C
282:B
255:2
251:S
225:C
219:B
197:C
194:,
191:B
188:,
185:A
163:C
141:B
138:,
135:A
111:2
107:S
77:3
72:R
45:2
41:S
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