Talk:Vitali set

Lebesgue measure

Why must Lebesgue measure of Vitali set be larger than one? If we consider something like Vitali set for interval [0,0.5] its measure will be less than (or equal to) 0.5 Then this set will be also enough for interval [0,1], because we can simply add 0.5 (which is rational) for each real in [0,0.5]. Then we can add 1, 2, 3,... and we will have every real in it. Set from beginning is true Vitali set. We can do similarly for interval [0,0.25], [0,0.125], etc. so I consider that there are Vitali sets with arbitrary small measure, and possibly also 0. Then infinite sum argument won't be enough to prove non-measurability, because 0+0+0+0+...=0, which is finite — Preceding unsigned comment added by Wojowu (talk • contribs) 17:30, 6 January 2012 (UTC)

The Lebesgue measure of countably many different Vitali sets must be larger than one. But the Lebesque measure of each of those different Vitali sets must be identical because or translational invariance. Thats the problem. Neither zero nor something larger will work. YohanN7 (talk) 17:09, 6 September 2012 (UTC)

Minor Point

It would be nice not to try relying on things like physical intuition in the opening sentences explaining these ideas. Most particularly, I noticed some kind of phrase like "imagine metal rods with uniform density". I do not believe that type of phrase is appropriate to describe abstract general math results. — Preceding unsigned comment added by 71.201.95.139 (talk) 01:57, 8 December 2012 (UTC)

Agreed. Plus, I believe the section on measurable sets shouldn't even be there. It replicates (in a worse way) content from the Lebesgue Measure article, which is much more complete and contextualized. — Preceding unsigned comment added by Walwal20 (talk • contribs) 18:27, 10 April 2020 (UTC)

History?

The article doesn't mention any history about the concept. Was Lebesgue measure defined after vitali sets where found, or was lebesgue measure already defined and the vitaly sets where found? — Preceding unsigned comment added by 181.29.52.110 (talk) 02:50, 2 July 2015 (UTC)

The "construction"'section does not include an explicit construction

I'm not an expert in this area. I expected to see an explicit definition of a Vitali set in the "construction"section. Right now, there doesn't seem to be one. Grover cleveland (talk) 16:04, 26 October 2021 (UTC)

There is no "explicit definition" in the sense of a formula of first-order logic with one free variable, φ(x), such that provably in ZFC the set of all reals x such that φ(x) holds is a Vitali set.

(Just for completeness, there is a formula φ(x) such that ZFC does not refute that φ defines a Vitali set.)

But the word "construction" here is not intended to evoke constructive mathematics or anything like that. It's a construction in an informal sense — you do this, then you do that, then you do the other thing. At various points you make arbitrary choices. --Trovatore (talk) 17:47, 26 October 2021 (UTC)

Infinite parity function

The articles on the infinite parity function provides a construction that is more-or-less identical to the construction of the Vitali set. The difference is that it uses the dyadic rationals instead of the rationals in the construction. But, of course, the rationals and the dyadic rationals map to one-another (one-to-one, onto) using the Minkowski question mark function. I would add this to the article, except that this is extremely minor "original research" on my part (well, it seems plainly obvious to me, but hey...); that, and the fact that the construction in infinite parity function is muddled and unreferenced. So, at a minimum, that article should be fixed so that the construction there makes it clear that it is just the Vitali set.

To be more precise, the infinite parity function uses the Vitali-like construction to define an infinite parity function f (and there are uncountably many of these, since one is asked to choose a representative from each coset.) and then notes that $f^{-1}(0)$ is not a Borel set. Given the construction, it appears that $f^{-1}(0)$ contains countably many copies of the Vitali set, all of which are offset from each-other by some dyadic rational. Only half the dyadic rationals are used; the other half land in $f^{-1}(1)$. It seems to me that the Vitali set could also be defined in this way, thus showing that there are uncountably many different Vital sets. 67.198.37.16 (talk) 01:42, 28 November 2023 (UTC)

The Vitali-type construction also shows up in the "second proof" in the article on the Axiom of determinacy, except there, it takes the form of Cantor space (inifinite binary strings) mod finite binary strings. The Cantor space maps to the reals in the usual way, and the finite-length binary strings map to the dyadic rationals. So I've now seen the exact same construction in three different articles in 24 hours. Does it have a name? My personal neologism is "a Vitali-type construction", although it could be called "the Vitali-type construction", since it seems there is only one. 67.198.37.16 (talk) 06:16, 28 November 2023 (UTC)

New material on measure and AC

So AxelBoldt added some interesting material about needing AC to prove that there's a non-measurable set, with a quibble that you might not need it if reality somehow manages to split the thin difference between ZFC and ZFC+"there exists an inaccessible cardinal". I made some minor tweaks to it.

But while the material is interesting, I'm not sure it belongs in exactly this article, at least in this form. If I remember correctly (and I don't have an immediate cite for this), you don't need the inaccessible to prove the consistency of ZF+"every set of reals has the property of Baire", which also refutes the existence of a Vitali set. So at least the stuff about the inaccessible cardinal seems to be more about measurable sets than about the Vitali set per se.

I wonder if it would fit better at non-measurable set? --Trovatore (talk) 06:01, 29 December 2023 (UTC)

The material should definitely be moved to non-measurable set where it properly belongs; this article could then just give a very short summary and refer people there. The result you mentioned is also very interesting and should be mentioned here, since it implies that "existence of non-measurable sets" and "existence of Vitali sets" are not exactly equivalent. Cheers, AxelBoldt (talk) 21:15, 31 December 2023 (UTC)