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Math Help - Measurable Set

  1. #1
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    Measurable Set

    Let n_1,n_2,n_3,... be an increasing sequence of positive integers. Let E = \{x \in [0,2\pi] : lim_{k \to \infty} sin (n_k x)\} exists. Prove that E is Lebesgue measurable and evaluate m(E).

    Hey this is something really new. How do I approach the question? Have been thinking for quite long and still couldn't get an idea on how to start.

    Thank you in advance.
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  2. #2
    MHF Contributor
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    Re: Measurable Set

    Well, you are looking at a subset of the interval \[0,2\pi\]. What do you know about sin x in that interval? Where is it zero? Where is it one? Where is it somewhere in between? Let's try some points...

    x=0\text{ implies }sin(n_k x)=sin(0)=0. Therefore, 0 \in E. What other points are in there? For what points will that limit not exist? For instance, what is sin(\infty)? I guess that depends on if infinity is 2\pi periodic (hint: it usually is NOT. Unless it is 2\pi periodic for all finite examples, then it will NOT be 2\pi periodic in the limit.) Figure out when sin is constant for all increasing integers (like we did for 0), and you will find when its limit exists.
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