On the problem of measuring sets of points on a straight line, by Giuseppe Vitali

This is an English translation of Giuseppe Vitali’s paper of 1905 on the question of the measurability of certain sets by the standard rules of measure, given the axiom of choice/well-ordering: Sul problema della misura dei Gruppi di punti di una retta: Nota, published by Tip. Gamberini e Parmeggiani, 1905.

English translation by James R Meyer, copyright 2022 www.jamesrmeyer.com

On the problem of measuring sets of points on a straight line

by Giuseppe Vitali

The problem of measuring sets of points of a straight line r is that of determining for each set A of points of r a real and positive number η(A), which will call the measure of A, so that: (Footnote: See: Leçons sur l’intégration etc, by H. Lebesque p.103, Paris, Gauthier-Villars, 1904.)

(1) Two sets that can be made to coincide with a convenient rigid displacement of one of them have the same measure.

(2) The sum set of a finite number of sets, or of a denumerable infinity of sets, without points common to any two, has a measure that is the sum of the measures.

(3) The measure of the set of all points of the interval (0, 1) is 1.

Let x be a point of r. The points of r that differ from x by any positive, negative or null rational number form a denumerable set A_x. If A_x1 and A_x2 are two such sets, either they are without any common points or else they coincide.

Consider different A_x sets as elements that constitute a set H. If P is any point of r, there will exist one and only one element of H to which P belongs.

For each element α of H, consider a point P_α of the interval (0, ½) that belongs to α, and we denote by G₀ the set of points P_α. If then ρ is any rational number, we will indicate with G_ρ the set of points P_α - ρ .

The sets G_ρ that correspond to the different rational values ​​of ρ are such that no two have common points, they are also a denumerable infinity and must have the same measure, by (1) above.

The sets

G₀, G _¹ ⁄2, G _¹ ⁄3, G _¹ ⁄4, …

all fall within the interval (0, 1), hence their sum must have a measure m ≤ 1.

But it must be the case that

  m = μ(G₀) +   ∞ ∑  n = 2   G _¹ ⁄n  

    = Lim            n → ∞ n · μ(G₀)  

and therefore

μ(G₀ ) = 0

But then the sum of all G_ρ corresponding to the different rational values of ρ must also have zero measure. But this sum is the set of all the points of r and therefore it should have infinite measure.

This suffices to conclude that:

the problem of measuring the sets of points of a straight line is insurmountable.

Some objections might be raised with respect to the argument regarding the G₀ set. The argument can be considered to be perfectly justifiable if one assumes that the continuum can be well-ordered. If one does not want to make that assumption, the above result means that:

a solution to the problem of measuring sets of points of a straight line is not possible if the continuum is well-ordered.