Math 2.0

Measure and integration Construction of measures Counterexample

Hausdorff paradox

Builds on Axiom of choiceRotationCountable setSpecial orthogonal group

Counterexample

Let S2={x∈ℝ3:∥x∥=1}S^2 = \set{x \in \R^3 : \norm{x} = 1} be the unit sphere. Assume the axiom of choice. There are a countable set D⊆S2D \subseteq S^2 and pairwise disjoint sets AA, BB, CC with S2∖D=A∪B∪CS^2 \setminus D = A \cup B \cup C. They can be chosen so that the four sets AA, BB, CC and B∪CB \cup C are congruent to each other by rotations of ℝ3\R^3. Hence there is no finitely additive, rotation-invariant function μ\mu on all subsets of S2S^2 with values in [0,∞)[0, \infty) and μ(S2)=1\mu(S^2) = 1.

Remarks

The proof uses two rotations φ\varphi and ψ\psi in the special orthogonal group SO(3)SO(3) with φ2=ψ3=id⁡\varphi^2 = \psi^3 = \id. They can be chosen to generate a free product of a group of order 22 and a group of order 33. The set DD collects the fixed points of the non-identity elements of this group, and the group acts freely on the rest of the sphere. Choosing one point in each orbit transfers a paradoxical splitting of the group to the sphere. For the consequence, finite additivity and invariance give μ(A)=μ(B)=μ(C)=μ(B)+μ(C)\mu(A) = \mu(B) = \mu(C) = \mu(B) + \mu(C). So all three are 00, and μ(D)=1\mu(D) = 1. But for a countable DD there is a rotation ρ\rho such that the sets D,ρ(D),ρ2(D),…D, \rho(D), \rho^2(D), \ldots are pairwise disjoint, which forces μ(D)≤1/n\mu(D) \le 1/n for every nn. Hausdorff used this to show that the measure problem has a negative answer in dimension 33 and higher; the Banach–Tarski paradox later strengthened the result.

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To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Countable set, concept, no evidence levelCountable setconceptElement, definition, no evidence levelElementdefinitionIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionProposition, concept, no evidence levelPropositionconceptRotation, concept, no evidence levelRotationconceptVector space, definition, no evidence levelVector spacedefinitionSet, definition, no evidence levelSetdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionLinear map, definition, no evidence levelLinear mapdefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionSubset, definition, no evidence levelSubsetdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionBinary relation, definition, no evidence levelBinary relationdefinitionDeterminant, definition, no evidence levelDeterminantdefinitionFunction, definition, no evidence levelFunctiondefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectChoice function, definition, no evidence levelChoice functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementHausdorff paradox, counterexample, no evidence levelHausdorff paradoxcounterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Countable set concept
  2. Element definition
  3. Identity matrix object
  4. Matrix definition
  5. Proposition concept
  6. Rotation concept
  7. Vector space definition
  8. Set definition
  9. Square matrix definition
  10. Transpose definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Linear map definition
  15. Family of sets definition
  16. Ordered pair definition
  17. Union definition
  18. Orthogonal matrix definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Multilinear map definition
  22. Cartesian product definition
  23. Union of a family of sets definition
  24. Subset definition
  25. Orthogonal group object
  26. Uniqueness quantifier definition
  27. Alternating multilinear map definition
  28. Binary relation definition
  29. Determinant definition
  30. Function definition
  31. Special orthogonal group object
  32. Choice function definition
  33. Axiom of choice statement
  34. Hausdorff paradox counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_paradox
About this page
Tag9G9G8ED97D
URIm2://9G9G8ED97D
Kindcounterexample
WritingW0 stub
MSC28A12, 03E25, 43A07
UsesAxiom of choice, Rotation, Countable set, Special orthogonal group
Same as
Changed2 November 2026

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