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I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:

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 would not depend on the axiom of choice.

Seriously? Lebesgue thought it should be possible to find non-Lebesgue-measurable sets without using choice? Since he had a record of opposing the use of the axiom of choice, wouldn't it make more sense for him to expect all "tame sets" (sets that don't involve $\mathsf{AC}$) were measurable? Indeed, Lebesgue went on to studying properties of Borel sets in detail (see Sur les fonctions représentables analytiquement), presumably to establish a framework of analysis that does not involve pathological things like Vitali sets.

So my questions are:

  1. Did Lebesgue ever change his mind and expect all "tame sets" to be measurable?
  2. If not, who first suggested that $\mathsf{ZF}$+(some weak choice principle)+"all sets are measurable" might be consistent?
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    $\begingroup$ I think it's easy to forget in hindsight that there's really no reason a priori to expect that you're going to be able to assign measures to arbitrary sets of reals (or even arbitrary 'tame' sets for a sufficient broad notion of tameness). It's easy to construct examples of non­–Riemann integrable functions and non–Jordan measurable sets. Lebesgue's approach was at the time just the best one that had been developed and it would be easy for someone at the time to imagine that there there might be a better one. $\endgroup$
    – James E Hanson
    Commented Sep 12 at 18:52
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    $\begingroup$ I think the trouble comes from the word "wanted". As far as I know, Lebesgue did think that you couldn't get a Lebesgue unmeasurable set without AC, but knew he hadn't proved it. So only a Lebesgue unmeasurable set constructed without AC would contradict his hypothesis, and therefore he "wanted" it as a precondition for changing his prior belief. In any case, it seems nobody predicted the relationship with (strongly) inaccessible cardinals, shown by Solovay and Shelah. $\endgroup$
    – Robert Furber
    Commented Sep 12 at 20:48
  • $\begingroup$ hsm.stackexchange.com $\endgroup$
    – Najib Idrissi
    Commented Sep 13 at 9:01
  • $\begingroup$ @RobertFurber Any reference for Lebesgue's view? $\endgroup$
    – new account
    Commented Sep 13 at 12:56
  • $\begingroup$ The main resource I have on Lebesgue's view on these things is his letter to Borel, part of a discussion that Borel started about Zermelo's paper that used the axiom of choice to prove the well-ordering theorem. Lebesgue says he proved the existence of a Lebesgue measurable set that is not a Borel set in his thesis but he doubts that such a set could be "named". He also says that the existence of infinite Dedekind-finite sets has not been ruled out. $\endgroup$
    – Robert Furber
    Commented Sep 24 at 12:48

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