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A proof of an area as a set function

  1. Nov 16, 2009 #1
    Hi guys, recently, I am a freshmen majoring in Physics, recently using Apostle Calculus to self-study. However, I am having a hard time knowing whether if my proof is correct or not, since there isn't any solution.
    1. The problem statement, all variables and given/known data
    Prove that following set is measurable and has zero area: a set consisting of a single point.

    3. The attempt at a solution

    First, I have considered the Axiom of Choice of scale{Every rectangle R is in Measurable Set. If the edges of R have lengths h and k then a(R)=hk} to show that h and k both is zero therefore, a(Point)=0

    but I couldn't convince myself that a point is an rectangle!! So my former approach is not proper.

    and I couldn't think up any other approaches so far.

    would anyone be king enough to give me a few suggestions?

    many thanks!!
     
  2. jcsd
  3. Nov 16, 2009 #2

    Office_Shredder

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    Think about making really small rectangles around the point
     
  4. Nov 29, 2009 #3
    Thank you very much indeed!!

    so here is my new approach, what would the problem be in my proof? :
    let there be a point in a rectangle [itex]a^2[/itex]
    [tex]\exists a\in R[/tex]
    s.t. for all [itex]x\in R, 0<a<x[/itex]
    [tex]\implies a^2=0[/tex]
    [tex]\implies f(point)=0[/tex]
     
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