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Prove if S and T are sets with outer content zero, SUT has outer content zero.

  1. Nov 9, 2012 #1

    Zondrina

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    1. The problem statement, all variables and given/known data

    Suppose that S and T are sets with outer content 0, prove that SUT also has outer content zero.

    2. Relevant equations

    C(S) denotes the outer content.

    C(S) = C(T) = 0

    Also : [itex]C(S) = inf \left\{{ \sum_{k=0}^{n} A_k}\right\}[/itex] where Ak is the area of one of the sub-rectangles Rk.

    3. The attempt at a solution

    So we want to show that C(SUT) = 0 using the fact C(S) = C(T) = 0. I'm not really sure where to start this one though. First time I've seen anything like it and a quick search yielded no results about outer content at all.

    I do have one theorem though. If S is a curve of finite length L, then C(S) = 0. I also figured ( not positive about this ) that C(∅) = 0.
     
    Last edited: Nov 9, 2012
  2. jcsd
  3. Nov 9, 2012 #2
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  4. Nov 9, 2012 #3

    Zondrina

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    Hmm I sort of see what you're saying. I'm confused as to why you took your sums and unions up to ∞ rather than to n and then later argued as n → ∞, C(S) or C(T) → 0. So given any positive ε :

    We take a set of rectangles R'k such that [itex]T \subseteq \bigcup_{k=1}^{n} R_{k}^{'}[/itex] and if I sum all the rectangles up to n, it will be smaller than (1/2)ε.

    We take another set of rectangles R''k such that [itex]S \subseteq \bigcup_{k=1}^{n} R_{k}^{''}[/itex] and if we sum all these rectangles up to n it will also be smaller than (1/2)ε.

    So hopefully I'm not mistaken here, but you asked me to consider the union of all the rectangles together.

    So we take a set of rectangles Rk such that [itex]S \cup T \subseteq \bigcup_{k=1}^{n} {R_{k}^{'}} \cup {R_{k}^{''}}[/itex] and if we sum all these rectangles, it will be less than (1/2)ε + (1/2)ε = ε.
     
  5. Nov 9, 2012 #4
    Yes this what i ment.
     
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