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Basic measure theory

  1. Feb 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that if [itex]E \subset B[/itex] and [itex]B \in L(\mathbb{R})[/itex] (where L(R) denotes the family of Lebesgue measurable sets on the reals) with [itex]m(B) < \inf [/itex], then [itex]E \in L(\mathbb{R})[/itex] if and only if [itex]m(B) = m^{*}(E) + m^{*}(B - E)[/itex], where [itex]m^*[/itex] denotes the Lebesgue outer measure.

    2. Relevant equations
    Basic set theoretic manipulations.

    3. The attempt at a solution
    The forward direction follows by the definition of the outer measure. As for the reverse direction, we need to show that for any set A contained in the reals,
    [tex]m(A) \geq m^{*}(A \cap E) + m^{*}(A - E)[/tex]
    (the less than or equal direction follows from subaddivity).
    I'm not really stuck at this point, but I'm wondering about my approach from here. I simply considered the three cases where A contains B, A is contained in B and contains E, and A is contained in E. I'm fairly certain this works out in each case as it should. This seems like a pretty straightforward approach, but is there a better way? I understand this isn't exactly the most interesting of problems but it helps me to know what other tools are available to me at this point. Thanks in advance.
  2. jcsd
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