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Lebesgue measure

  1. Aug 16, 2007 #1
    1. The problem statement, all variables and given/known data
    m((a,b])=b-a is defined as the lebesuge measure

    what is m([a,b))?





    3. The attempt at a solution
    m({a})=0 for any a in R?

    so m([a,b))=m((a,b])?
     
  2. jcsd
  3. Aug 16, 2007 #2

    Dick

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    Sure.........
     
  4. Aug 16, 2007 #3

    HallsofIvy

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    If this is a course in measure theory, I'm sure the problems will get harder!
     
  5. Aug 16, 2007 #4
    They didn't specify that m({a})=0 for any a in R. So it was more a problem, of ambiguity.

    It was part of a bigger problem.
     
  6. Aug 17, 2007 #5

    matt grime

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    They did specify that m(pt)=0 - it is deducible from your first post. A point pt lies in any interval (pt -e/2 , pt+e/2] for any e, hence m(pt)<e for all e, thus it is zero.
     
  7. Aug 18, 2007 #6
    Good point or maybe more easily it can be worked out from letting b=a

    m((a,b])=b-a is defined as the lebesuge measure

    => m((a,a])=a-a=0
    => m(pt)=0
     
  8. Aug 18, 2007 #7

    matt grime

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    (a.a] does not equal the set {a}. So what the measure of the (empty) set (a,a] is does not tell you what the measure of the non-empty set {a} is. (Even assuming that a one point set is measurable, of course.)
     
    Last edited: Aug 18, 2007
  9. Aug 18, 2007 #8
    (a,a] dosen't make sense does it. It should be lim n->infinity(a-1/n,a]={a}
     
  10. Aug 18, 2007 #9

    matt grime

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    Taking limits of sets needs some careful consideration. Do you mean direct or inverse limit? It's the inverse limit, by the way.

    If I were you I'd not attempt to write things like: the limit of these sets is that set. Stick to sequences of numbers, not sequences of sets.
     
  11. Sep 10, 2007 #10
    Measure of an interval

    1.
    Ans: still b-a. ...........
     
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