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Strange thing

  1. Sep 3, 2008 #1
    I realized something weird.
    That, suppose you take the rationals in [0,1], call this set [tex]Q.[/tex] [tex]Q[/tex]'s a Borel set, so if [tex]\mu[/tex] is Lebesgue measure, [tex]\mu(Q)=inf(\mu(V), V [/tex]open,[tex] Q \subset V).[/tex]
    [tex]Q[/tex] can be covered by open sets of total measure [tex]\le 1[/tex] by counting the rationals; cover the first rational by an interval size 1/2, the second by an interval size 1/4 ...
    But, [tex]Q[/tex] can also be covered by open sets of total measure [tex]\le 1/2[/tex] in the same way. Or by an open covering of arbitrarily small total measure ...
    It's strange considering that the rationals are dense in the irrationals. Yet you could leave 999/1000 of the irrationals out of the open cover ...
  2. jcsd
  3. Sep 4, 2008 #2


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    Hi Laura! :smile:

    Why is that strange?

    Measure is supposed to be like weighing …

    if you tipped all the rationals into a pan and weighed them, you wouldn't expect them to weigh anything, would you? :wink:
  4. Sep 4, 2008 #3
    So you're surrounding each rational in [0,1] by an open interval; the rationals are dense in [0,1]; yet it can be arranged so practically all (say 999999 out of a million) of the irrationals are not covered by one of the open intervals. That's what is weird.
  5. Sep 4, 2008 #4


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    This observation (rationals are a set of measure zero) is one way of showing that the set of real numbers between 0 and 1 is uncountable. This is useful for those who don't like Cantor's diagonal proof.
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