Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Vitali set

  1. Sep 3, 2011 #1
    Hello all,

    I have a doubt on the Vitali set.
    In its construction we define equivalence classes, two numbers belonging to one if their difference is rational.
    Then we "pick" a member per class, forming a set. It is then shown that the interval [0,1] is a disjoint union of such set (after a rational translation), hence after some reasonsing this set is non-measurable.
    It seems to me that one such set is formed by a rational number plus all irrationals.
    Now, the interval [0,1] is also the disjoint union of the rationals and the irrationals.
    Hence, as the rationals are a set of zero lebesgue meausure, and as the irrationals are not measurable (differing by only one member from the the set built during the Vitali construction), one could conclude the interval [0,1] is the disjoiunt union of a zero measure set and a non - measurable set, which seems a contradiction.
    Can anybody point out my mistake as I can not?


  2. jcsd
  3. Sep 3, 2011 #2


    User Avatar
    Science Advisor

    No, there is no such set containing all irrationals. For example, [itex]e- \pi[/itex] is not a rational number so [itex]e[/itex] and [itex]\pi[/itex] cannot be in the same equivalence class.

    Your error, as I said, is in thinking that there was such a set containing all irrational numbers.
    The set of all irrational numbers, between 0 and 1, is not, of course, "non-measurable"- it has measure 1.
  4. Sep 3, 2011 #3


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    This is where you went wrong.
  5. Sep 4, 2011 #4
    Correct indeed, many thanks
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook