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Math Help - Tow problems in measure need to be solve

  1. #1
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    Tow problems in measure need to be solve

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  2. #2
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    What is your definition of measurable? I assume you mean that
    f^{-1}(B) \in \mathcal{F}
    for each Borel set.

    So here are some hints;
    First show 2) by considering sets of the form [a,\infty) then show 1 by using the 2 complement and sets
    <br />
\{f \geq n\}

    For 3 consider sets of the form  [a-1/n,a+1/n].

    Q2 is wrong, you need the function to be surjective. You should attempt it first (and post here).
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  3. #3
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    Let f be measurable function if for each real no. a the set {x in E : f(x)>a} is measurable.
    For all real a, the set { x in E :f(x)≥a} is measurable since,
    {x in E : f(x)≥a } = intersection { x in E∶ f(x)>a-1/n }
    = a measurable set and hence {x in E : f(x)=a}
    = {x in E : f(x)≥a }-{x in E : f(x)>a} is measurable.

    Also {x in E : f(x)=∞} = intersection { x in E: f(x)>n} is measurable.

    {x in E : f(x)<a} is measurable as it is the complement of 1. Clearly the set {x in E : |f(x)|< a} is measurable for two measurable sets {x in E : f(x)>a} and {x in E : f(x)<a}.
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