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Math Help - measure theory intergration

  1. #1
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    measure theory intergration

    The question is:
    Show that if  g:\mathbb{R} \longrightarrow [0,\infty) equals 0 outside a bounded interval and  \int g^2 < \infty , then  \int g < \infty

    It doesn't appear so difficult, but i dont know how to progress in it. I say let I be the bounded interval, then  \int g^2 = \int_{\mathbb{R}} g^2 \chi_I < \infty how do i progress from here? i dont know how to get  \int g from this! thanks
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  2. #2
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    Divide I=A\cup B, A=\{x\in I: |f(x)|\leq 1\} and B:=I\setminus A. Both sets are measurable. Show that f is integrable both in A and B because it is dominated by the identically 1 function in A (which is of finite measure) and by f^2 in B, integrable by hypothesis.
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  3. #3
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    Quote Originally Posted by Enrique2 View Post
    Divide I=A\cup B, A=\{x\in I: |f(x)|\leq 1\} and B:=I\setminus A. Both sets are measurable. Show that f is integrable both in A and B because it is dominated by the identically 1 function in A (which is of finite measure) and by f^2 in B, integrable by hypothesis.
    Thanks a lot!!!!!!! I wouldn't have ever thought of dividing I!!!
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