Talk:Non-measurable set

message: Why does the opening sentence restrict us to finite measures?  There are certainly nonmeasurable subsets of the real line which are contained in no set of finite measure.  —Preceding unsigned comment added by 68.185.170.23 (talk) 04:29, 28 January 2009 (UTC)
replies: text: I agree.  I see no reason why this restriction is made.  I think it should read "In mathematics, a non-measurable set is a subset of a set with non-zero measure...", since any subset of a set with zero measure is defined as measurable (at least in Lebesgue measure). --67.193.128.233 (talk) 18:12, 8 March 2009 (UTC)
thread_title: Opening sentence

message: quote : ...Vitali's theorem which basically states that you can take an interval of length 1, dissect it into pieces, move the pieces around and get an interval of length 2 (sometimes this result is called the Hausdorff paradox). 
This is nonsense, you cannot get an interval of length 2. For dimensions 1 and 2 Banach proved existence of finitely additive translation invariant measures. He actually constructed them. Whoever wrote this should actually learn what Hausdorff-Tarsky Paradox is about. It is in R^3 !  Vitali's set is NEVER called the Hausdorff paradox. 
—The preceding unsigned comment was added by 206.188.85.61 (talk) 19:04, 2 May 2007 (UTC)
thread_title: Nonsense in the article

message: "The measurable sets on the line are formed by countable unions and intersections of intervals"
AFAIK these are Borel-sets, while the article talks about Lebesgue non-measurable sets (the Borel non-measurable sets aren't that interesting because their existence is trivial and is not "a source of great controversy"). bungalo (talk) 10:09, 17 November 2010 (UTC)
replies: text: I am not sure that the lead of this article (that "gives a general overview of the concept") must be so exact. Anyway, I have added the missing reservations. Boris Tsirelson (talk) 14:27, 17 November 2010 (UTC)
replies: text: Well, really, that's the problem with this sort of article.  To be honest I think this article should not exist.  It should be merged somewhere.  Wikipedia should never involve itself in "lies to children". --Trovatore (talk) 19:43, 17 November 2010 (UTC)
replies: text: That is, Wikipedia should include some scientific literature, but not some popular science. Really? Why? Boris Tsirelson (talk) 22:06, 17 November 2010 (UTC)
replies: text: There is nothing wrong with trying to make things clear to readers of limited background.  We just can't permit that to be an excuse for saying things that aren't true. --Trovatore (talk) 22:12, 17 November 2010 (UTC)
thread_title:  Borel or Lebesgue? 

message: There is perpetual confusion between saying the negation of X is consistent, and jumping to saying it is true.  The joker is, of course, the need for Solovay to assume the existence of an inaccessible cardinal...  this article needs drastic rewriting in this regard.  — Preceding unsigned comment added by 98.109.239.253 (talk) 03:24, 5 February 2012 (UTC)
replies: text: I'm not sure I understand your complaint.  I don't see anywhere in the article that claims it is true that all sets are measurable (I'm guessing that's your "negation of X"?), only that it's true in the Solovay model.  And yes, if there's no inaccessible, then there's no Solovay model.  So what? --Trovatore (talk) 20:13, 5 February 2012 (UTC)

text: Oh wait a minute, maybe I see.  Is your complaint that claim that if the existence of an inaccessible is consistent, then the Solovay model exists?  I have to agree that that is suboptimally worded.  In context, the point is that if it's consistent that there's an inaccessible, then it's consistent that all sets of reals are measurable.  It follows that there's a model that satisfies "all sets of reals are measurable".  However, it doesn't follow that that model is the Solovay model.  I kind of think "more nonsense" is putting things a little strongly, but it's true that the text is not quite correct as worded.  Any suggestions for fixing it while still getting the point across, and hopefully without overly complicating the text? --Trovatore (talk) 21:21, 5 February 2012 (UTC)
thread_title: More nonsense

message: In the example of the unit circle, it says "the circle gets partitioned into a countable collection of disjoint sets." Aren't there uncountably many disjoint orbits? I see the same thing on the axiom of choice page.
173.25.54.191 (talk) 22:48, 19 November 2012 (UTC)
replies: text: Yes, there are uncountably many disjoint orbits (each of which is countable).  But that's not the partition the text is talking about.  If you have a set that picks out exactly one point from each orbit, then the collection of all translates of that set is a partition into a countable collection of disjoint sets.  Maybe this point can be stated more clearly; any suggestions? --Trovatore (talk) 23:09, 19 November 2012 (UTC)

text: Looks fine.  Except, is everyone going to understand what translate means as a noun?  Might have to think about that... --Trovatore (talk) 02:27, 20 November 2012 (UTC)
thread_title:  countability in example 

message: The article says
:Consider the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable (more specifically, G is isomorphic to \mathbb{Q}/\mathbb{Z}) while S is uncountable.
My question is how \mathbb{Q}/\mathbb{Z} can be a group? What is its identity element and group operation then? And what is the isomorphism between these sets? I believe the author meant \mathbb{Q}/\{0\} (or rather  \mathbb{Q}\setminus\{0\}). I am right?
94.193.53.206 (talk) 11:26, 24 January 2013 (UTC)
replies: text: The quotient of any abelian group G by its subgroup H has a canonical structure of a group, and Q/Z is no different.  The identity element is still 0 and the group operation is addition.  On the unit circle in C, the group operation is multiplication.  The isomorphism is given by the exponential function e^{ix} (up to adjusting the constant in the exponent). Tkuvho (talk) 15:14, 24 January 2013 (UTC)
thread_title:  Example