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Math Help - Measurable Function

  1. #1
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    Measurable Function

    i couldn't find a way to show this function is measurable.

    f(x)=1 if x \epsilon F and f(x)=sin(x) if x\varepsilon [0,1]\F where F is the Cantor Set.

    i will be appreciate for any help.

    edit: i defined a function g(x)=sin(x) over the interval [0,1] but i am not sure about f \approx g because f takes value 1 for some points in [0,1] so there will be discontinuity
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  2. #2
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    Quote Originally Posted by gilgames View Post
    i couldn't find a way to show this function is measurable.

    f(x)=1 if x \epsilon F and f(x)=sin(x) if x\varepsilon [0,1]\F where F is the Cantor Set.

    i will be appreciate for any help.

    edit: i defined a function g(x)=sin(x) over the interval [0,1] but i am not sure about f \approx g because f takes value 1 for some points in [0,1] so there will be discontinuity
    The Cantor set has measure 0. Thus the functions f and g are equal almost everywhere. It follows that if one is measurable then so is the other.
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  3. #3
    Super Member girdav's Avatar
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    Write f(x) = \mathbf{1}_F(x)+\sin x\mathbf{1}_{\left[0,1\right]\setminus F}(x). Because F is measurable, x\mapsto \mathbf{1}_F(x) and \mathbf{1}_{\left[0,1\right]F}(x) are measurable. x\mapsto \sin x is measurable because it's continuous.
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