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Real Analysis-Monotonicity

  1. Sep 18, 2014 #1
    **Moderator: Please move my thread to the correct section. Sorry about posting it in the wrong location**


    In example 2 on page 11 of the text I linked above, the book uses the monotonicity property of the outer measure to conclude
    ##m_*(Q) \le |Q|## where Q is a cube.
    However, in example 4 on the next page they require a more complicated argument to reach the conclusion
    ##m_*(R) \le |R|## where R is a rectangle.
    My question is, why couldn't they use monotonicity again?
    Last edited: Sep 18, 2014
  2. jcsd
  3. Sep 19, 2014 #2


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    The text is confusing!
  4. Sep 23, 2014 #3
    They define the exterior measure using coverings by closed cubes. Example 2 is an closed cube, and Example 4 is a rectangle, and that's what accounts for the difference. In fact, monotonicity has nothing to do with the inequalities you are worried about.
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