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Lebesgue measure

  1. Sep 17, 2009 #1
    I've seen the definition that an outer Lebesgue measure is defined as
    [tex]m_n^* (A) = \inf \left\{ \sum_{k=1}^{\infty} l(I_k) \, \left| \, A \subset \bigcup_{k=1}^{\infty} I_k \right}[/tex]
    where Ik are n-dimensional intervals and l(Ik) is the geometric length.

    It is not actually clear to me if A has to be a proper subset. That is, does
    [tex]A \subset \bigcup_{k=1}^{\infty} I_k[/tex]
    actually mean
    [tex]A \subseteq \bigcup_{k=1}^{\infty} I_k[/tex]
    [tex]A \subsetneq \bigcup_{k=1}^{\infty} I_k[/tex]
  2. jcsd
  3. Sep 17, 2009 #2


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    Why do you think the distinction between those two cases is important? Show that the sum of l(I_k) can be made arbitrarily close to the sum of the lengths of intervals J_k where each I_k is a proper subset of J_k. That's a good exercise.
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