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/
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 Ax. If Ax1 and Ax2 are two such sets, either they are without any common points or else they coincide.
Consider different Ax 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 Pα of the interval (0, ½) that belongs to α, and we denote by G0 the set of points Pα. If then ρ is any rational number, we will indicate with Gρ the set of points Pα - ρ .
The sets Gρ 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.
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)
μ(G0 ) = 0
But then the sum of all Gρ 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.