Felix Hausdorff
“Bemerkung über den Inhalt von Punktmengen”
(A Remark on the Measure of Point Sets)
• English Translation •

This is an English translation of Felix Hausdorff ’s Bemerkung über den Inhalt von Punktmengen, published in 1914 (Mathematische Annalen 75.3, 1914, pp.428-433), also commonly referred to as “The Hausdorff Paradox”. You can view the German original at Zenodo: Bemerkung über den Inhalt von Punktmengen.

English translation by James R Meyer, copyright 2025 jamesrmeyer.com

Translator’s note: I have changed the symbol for measure from Hausdorff ’s f to the symbol now commonly used μ, and changed Hausdorff ’s + symbol to ∪ where it indicates set union.

As is well known, E. Borel and H. Lebesgue attempted to treat the problem of determining the measure of sets of points axiomatically, at least to a certain extent. They postulated assigning a non-negative number μ(A) as the measure of every bounded set A of the n-dimensional Euclidean space E_n with the following conditions:

The constructive definition of measure given by Lebesgue does indeed fulfill these four requirements, but it does not assign measure to all (bounded) sets, but only to those that are “measurable”. The first example of a non-measurable set in the Lebesgue sense is given by G. Vitali. See A. Schoenflies, “Entwickelung der Mengenlehre”, Leipzig and Berlin, 1913, p.374, where, in addition to further examples from E. B. van Vleck and Lebesgue, the text that follows is also communicated by me. Incidentally, the system of non-measurable sets has the same cardinality as that of measurable sets and that of all sets. The question thus remains open whether the measure problem posed by requirements (α) to (δ) is solvable at all when extended to all bounded sets.

It is immediately apparent that a solution to the measure problem for E_n + 1 would also yield one for E_n (by assigning to an n-dimensional set A_n the (n + 1)-dimensional cylindrical body A_n + 1 with base A_n and height 1 and setting μ_n(A_n) = μ_n + 1(A_n + 1) ). A similar case is the n-dimensional spherical space K_n, where condition (α) is to be replaced by the condition that K_n itself should have a positive measure, say 1. Lastly, the measure determinations for the straight line E₁ and the circle K₁ are identical problems.

We first show that, given conditions (α) to (δ), a determination of measure is not possible by any means; as already noted, it suffices to prove this for the circle. For this purpose, we divide the circle’s periphery K into countable congruent sets. If we take the radius = 1/2π, i.e., μ(K) = 1, represent the points of the circle by a real variable x such that two values with an integer difference correspond to the same point on the circle, and understand δ to be an irrational number, then for each point x there belongs a countable set:

P_x = {…, x − 2δ, x − δ, x, x + δ, x + 2δ, …}

and two sets P_x, P_y, if they are not identical, have no points in common. If we now select exactly one point x from every set P_x, we obtain a set:

A₀ = {x, y, z, …}

which, by rotation by an arbitrary multiple mδ of δ, becomes the set:

A_m = {x + mδ, y + mδ, z + mδ, …}

Two such sets with different indices have no point in common, and:

K = … ∪ A_-1 ∪ A_-2 ∪ A₀ ∪ A₁ ∪ A₂ ∪ …

All these sets are congruent, i.e: they must have the same measure μ(A_m ) = a. Since K includes the subset A₁ ∪ A₂ ∪ … ∪ A_n , then 1 ≥ na must be true for every natural number n, and so a = 0, and thus the condition (δ) is violated. It follows incidentally that these sets A_m are not measurable in the Lebesgue sense (their outer measure is positive, their inner measure zero). But the essential point is that we have not used Lebesgue’s definition of measure at all, but only his postulates.

The contradiction that results concerns requirement (δ), which, however, constitutes precisely the specific advantage of the Borel-Lebesgue measure theory over the older Peano-Jordan measure theory. Let us, however, abandon this requirement and inquire about a possible solution to the measure problem by restricting ourselves to the postulates (α) , (β),  (γ), which presumably represent the minimum of what must be required of the measure of the point sets.

Even under these conditions, the problem is unsolvable, at least for the sphere and therefore for three or more-dimensional Euclidean space. For this conclusion, we will show that a half of the sphere can be congruent with a third of the sphere, or more precisely, that (apart from a countable set) the sphere K can be partitioned into three sets A, B, C such that A, B, C and B ∪ C are pairwise congruent.

Let φ be a half-rotation (by π) and ψ a third-rotation (by 2π/3) about an axis different from the first. Since φ² = ψ³ = 1, these two rotations generate a group that can initially be represented as follows:

(G)   1 | φ, ψ, ψ² | φ ψ, φ ψ², ψ φ, ψ²φ | …

in which we have separated the products of different numbers of factors with vertical lines; here φ, ψ, ψ² are considered basic factors. The general form of a product of two or more factors is one of the following four cases:

α = φ ψ^m₁ φ ψ^m₂ … φ ψ^m_n,

β = ψ^m₁ φ ψ^m₂φ … ψ^m_nφ,

γ = φ ψ^m₁ φ ψ^m₂ … φ ψ^m_n φ,

δ = ψ^m₁ φ ψ^m₂… φ ψ^m_n.

where n is a natural number and m_i = 1 or 2.

With a suitable choice of the axes of rotation, no product of one or more factors is equal to the identity 1, and hence the rotations written in the form (G) are pairwise distinct. To show this, we first note that a product which is 1 can always be assumed to be of the form α, since from β = 1 it would follow that φ β φ = α = 1. Similarly from γ = 1 we have φ γ φ = δ = 1, and lastly from δ = 1 we have:

ψ^3-m₁ δ ψ^m₁ = α′ = 1

The orthogonal transformations corresponding to our rotations, if we set the ψ-axis as the z-axis, and we set the φ-axis in the x z-plane, and we denote the angle between both axes as ½ θ, have the following form:

(ψ)		x′	=	x λ	−	y μ
		y′	=	x μ	+	y λ
		z′	=	z

(φ )		x′	=	− x cos θ	+	z sin θ
		y′	=	− y
		z′	=	− x sin θ	+	z cos θ

(φψ )		x′	=	− x λ cos θ	+	y μ	+	z λ sin θ
		y′	=	− x μ cos θ	−	y λ	+	z μ sin θ
		z′	=	x sin θ			+	z sin θ

where λ = cos ^2π⁄₃ = ⁻¹⁄₂ and μ = sin ^2π⁄₃ = ^√3⁄₂ .

The transformation from ψ to ψ² is effected by replacing μ with −μ.

Let α be a product of n double factors φ ψ or φ ψ², α′ one of n + 1 such, i.e., α′ = α φ ψ or α′ = α φ ψ². Let α transform the point 0, 0, 1 into the point x, y, z and α′ into x′, y′, z′, such that between these coordinates exist the equations φ ψ or the equations φ ψ² resulting from them by interchanging μ with −μ. We claim that x, y, z are of the form:

x = sin θ (a cos θ ^n-1 + …),

y = sin θ (b cos θ ^n-1 + …),

z = c cos θ ⁿ + …

where x and y are products of sin θ with a polynomial (n - 1)^th degree, and z a polynomial of n^th degree in cos θ. In fact, this is true for n = 1 since the point 0, 0, 1 is rotated through φ ψ or φ ψ² into λ sin θ, ± μ sin  θ, cos  θ and passes from degree n to n + 1, giving:

x′ = sin θ (a′ cos θ ⁿ + …),

y′ = sin θ (b′ cos θ ⁿ + …),

z′ = c′ cos θ ^n + 1 + …

where we also have:

a′ = λ (c − a),

b′ = ± μ (c − a),

c′ = c − a,

c′ − a′ = (1 − λ) (c − a) = ³⁄₂( c− a)

By repeatedly applying this recursion formula it follows that after n rotations φ ψ or φ ψ² the point 0,0,1 has passed into x, y, z where:

z = (³⁄₂)^n-1 cos θ ⁿ + …

in any case not identically reduced to 1. The product a can therefore only be equal to identity 1 for a finite number of particular values of cos θ, and it is therefore possible, by avoiding no more than a countable set of values, to choose θ such that no product α becomes equal to 1.

Since θ is chosen in this way, the transformations of (G), except for the first one, each have only two fixed points, which together form a countable set Q; if we set the entire sphere K = P ∪ Q, then, according to our intended determination of the area, μ(Q) = 0 and μ(P) = μ(K) > 0 must apply (By a suitable rotation, whose axis does not pass through any point of Q and whose angle is not equivalent to any of the countably many differences in value between the points of Q, Q becomes a congruent subset of P, and by repeated application, it becomes clear that, for every natural number n, μ(P) ≥ nμ(Q), and therefore μ(Q) = 0). If we now consider only the set P, then each of its points x corresponds to the countable set P_x of pairwise distinct points:

x, x φ, x ψ, x ψ², x φ ψ, …

into which x is transformed by the transformations of (G); if we select exactly one point from each set P_x then a set M is created:

M = {x, y, z,…}

and:

P = M ∪ Mφ ∪ Mψ ∪ Mψ² ∪ Mφ ψ ∪ …

Finally we now assert: it is possible to distribute these sets, or their corresponding transformations, among three classes A, B, C such that:

(1) of two transformations ρ, ρ φ one belongs to A and the other to B ∪ C.

(2) of three transformations ρ, ρ ψ, ρ ψ² one belongs to each of A, B, and C.

Suppose, in fact, that the products of n or fewer factors are already distributed such that these conditions are satisfied, insofar as the transformations in question occur among those already distributed. A product of n factors whose last factor is ψ or ψ² is called ψ_n, and a product whose last factor is φ is called φ_n. Every product of n + 1 factors is then of one of these three forms: ψ_nφ,  φ_nψ,  φ_nψ², and we distribute them according to the following:

1. If ψ_n belongs to A, B, C, then ψ_nφ belongs to B, A, A.
2. If φ_n belongs to A, B, C, then φ_nψ belongs to B, C, A, and φ_nψ² belongs to C, A, B.

This means that the products of n + 1 or fewer factors are distributed in the desired manner; at the same time, it now becomes clear what role the independence of φ and ψ performs, because if one product = 1 or two structurally different products were the same, the required distribution could lead to contradictions.

By placing the identity in class A, we indicate the beginning of the distribution as: Translator’s note: The top line in the row A indicates a product derived from the set B by the rules (i) and (ii) above, and the bottom line in the row A indicates a product derived from the set C by the rules (i) and (ii) above.

A	1		ψ φ,φ ψ² ψ²φ	φ ψ φ	ψ φ ψ φ,ψ²φ ψ φ,ψ²φ ψ² ψ φ ψ²φ,ψ²φ ψ²φ	…
B		ψ, φ		ψ φ ψ,ψ²φ ψ ,ψ²φ	φ ψ φ ψ	…
C		ψ²	φ ψ	ψ φ ψ²,ψ²φ ψ²	φ ψ φ ψ²,φ ψ²φ ψ	…

while the following two compilations show how, by condition (1), classes A and B ∪ C are related to each other, and by condition (2), classes A, B, C are related to each other:

Let us now also denote the unions of the respective sets by A, B, C:

A = M ∪ Mψ φ ∪ Mψ²φ ∪ Mφ ψ² ∪ …

B = Mψ ∪ Mφ ∪ …

C = Mψ² ∪ Mφ ψ ∪ …

and so:

Aφ = B ∪ C,  Aψ = B,  Aψ² = C

The sets A, B, C, B ∪ C are congruent and A would have to be a half and one-third of the sphere at the same time.

The problem itself of measure, even without Lebesgue’s requirement (δ), is therefore unsolvable for the sphere and for three-dimensional or more-dimensional space.

For the circle, the straight line, and the plane, the question of a content determination that satisfies the conditions (α),  (β),  (λ) must remain open, since the structure of the motion group in these cases does not permit the above procedure.

Greifswald, February 27, 1914.