Construction of nonmeasurable sets

Question

I have a history question for which I've had trouble finding a good answer.

The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et al. put the blame squarely on this axiom and its non-constructive character. It was noticed however that some amount of choice was required to get measure theory off the ground, namely Dependent Choice seemed to be the principle typically employed. But the full axiom of choice which allows uncountably many arbitrary choices to be made is of a different character, and is the culprit behind the pathological sets. This viewpoint was not really justified until Solovay showed in the 1960s that ZF+DC could not prove the existence of a nonmeasurable set, assuming the consistency of an inaccessible cardinal.

My question is, in the many years before Solovay's theorem, was there any effort aimed at showing the existence of a nonmeasurable set without the use of the full AC? Was something like the following question ever posed or worked on: "Can constructions similar to those of Vitali, Hausdorff, and Banach–Tarski be done without appeal to the Axiom of Choice?"

Answer 1

Paul Cohen posed the question of getting a model of "All Sets Lebesgue Measurable" in his early talks on his own results. (He did not mention the principle of Dependent Choices. Adding that to the problem was my idea.) I know of no work trying to prove the Vitali result constructively. Certainly Cohen's conjecture (which I presume was widely shared) was that the use of choice was essential.

It is quite striking (if one works through Halmos) that all the positive results in measure theory can be carried out in ZF + DC. Only the counterexample section uses full choice.

Answer 2

I would like to point out that the question on the existence of a non-measurable set without the use of the full AC was technically established in the literature before Solovay's result (which afaik goes back to March-July 1964).

In 1938 Sierpinski had established (cf. "Fonctions additives non complètement additives et fonctions non mesurables", Fund. Math.) that a non-measurable set could be constructed from the assumption (in modern terminology) that there is a prime ideal on the power set of the natural numbers extending the ideal of finite sets. He explains that the existence of such prime ideal was proven by Tarski (cf. "Une constribution à la théorie de la mesure", Fund. Math.) with the aid of the Axiom of Choice (he proved it by transfinite induction).

But it remained an open question, especially in the 50's after Henkin's results, whether the existence of such prime ideals in the power sets, or, more generally, in Boolean algebras, was or not weaker than the Axiom of Choice. This was eventually settled by Halpern, who proved that it was, and his results first appeared on his doctoral dissertation, submitted in the spring 1962.

Sierpinski construction is quite simple: define a function $f$ on a real number $x$ to be either $1$ or $0$, depending on whether the subset of ones in (the non integer part of) its dyadic expansion (choosing the finite development for the rationals) is or not in the prime ideal defined on the powerset of the natural numbers extending the ideal of finite sets. It follows that $f$ has arbitrarily small periods (all numbers of the form $2^{-n}$) and that $f(1-x)=1-f(x)$. From this and the fact that $f$ only takes the values $0$ and $1$ it is not hard to show that it cannot be measurable, and the preimage of $0$ provides a non-measurable set.

Answer 3

Here is an earlier effort of Sierpiński: Sur une propriété de la décomposition de M. Vitali, Mathematica 3, 30-32 (1930). He took "Vitali's Decomposition", that is, the family of cosets of $\mathbb{Q}$ in $\mathbb{R}$ and divided it into (unordered!) pairs $\{Q,-Q\}$ of symmetric sets (excluding the one symmetric coset: $\mathbb{Q}$ itself). Given a choice set $\mathcal{Q}$ for this family of pairs let $S=\bigcup\mathcal{Q}$. Then $S$ is non-measurable.

Thus you can get a non-measurable set from AC for pairs, or from the existence of a linear order on the power set of $\mathbb{R}$. He already mentioned the latter in 1917: Sur quelques problèmes qui impliquent non mesurables, C. R. 164, 882-884 (1917).