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?"

A radical approach to Lebesgue's theory of integration, claims (in p. 154) that "Vitali's nonmeasurable set, appearing less than a year later [than Zermelo's Well-ordering theorem], was greeted by Lebesgue and many others as an empty exercise. They wanted an example of a nonmeasurable set whose construction ould not depend on the axiom of choice." – Andrés E. Caicedo Aug 29 '14 at 1:12