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Question on rectifiable sets

  1. Aug 11, 2010 #1
    1. The problem statement, all variables and given/known data
    I am trying to work my way through Analysis on manifolds by Munkres. Question 14.5 has me stumped. Any hints on how to tackle it would be appreciated. The question is:

    Find a bounded closed set in [tex]\mathbb{R}[/tex] that is not rectifiable

    2. Relevant equations

    A subset S of [tex]\mathbb{R}[/tex] is rectifiable iff S is bounded and the boundary of S has measure zero.

    The boundary of an interval in [tex]\mathbb{R}[/tex] has measure zero.

    3. The attempt at a solution

    I think I need a closed set who's boundary does not have measure zero. I presume it has to be an uncountable union of intervals of some description, but I have no idea how to approach the construction of such a thing.
  2. jcsd
  3. Aug 11, 2010 #2


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    Hint: fat Cantor set. Look it up.
  4. Aug 11, 2010 #3

    Thank you. I should have thought of that, but only considered the standard Cantor set. Thanks again.
  5. Aug 11, 2010 #4


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    Wow! That was fast. I see that "a word to the wise is sufficient" is sometimes true.
  6. Aug 11, 2010 #5
    I have come across fat cantor sets before and my problem was that I could not think of a closed set whose boundary had positive measure. As soon as you gave the hint the rest followed and I felt like a fool. Oh well such is the learning process. Thanks again for the help, I had been stuck on that for a couple of days.
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