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Show that X ⊂ ℜn has measure 0 if and only if ε > 0

  1. Dec 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Please I need your help in this question. I don't know how to answer it.

    The question: Show that X ⊂ ℜn has measure 0 if and only if ε > 0 there exists an infinite sequence of balls

    B_i ={ x ∈ R^n| |x-a_i | < r_i} with ∑ r[itex]^{n}_{i}[/itex] < ε such that X ⊂ ∪ [itex]^{\infty}_{i =1}[/itex]B_i



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 3, 2011 #2

    gb7nash

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    Re: measures

    What have you tried?
     
  4. Dec 3, 2011 #3

    micromass

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    Re: measures

    Please post an attempt at the solution or this thread will be deleted.

    Also it might be necessary to define your terms. How did you define "measure 0" etc.
     
  5. Dec 3, 2011 #4
    Re: measures

    I said
    choose ε > 0, , for n = 1, i = 1, let a be the center of the ball and raduis r. if | x- a| < r with Ʃ r < ε such that X [itex]\subset[/itex] B[itex]_{1}[/itex]. and keep trying for n =2 and generalise it? this is my guess?
     
  6. Dec 3, 2011 #5

    micromass

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    Re: measures

    And how did you define "measure 0"??
     
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