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The ingerals of a function on two different measures

  1. Feb 28, 2010 #1


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    Is the integral of a strictly positive function on a set of positive measure strictly positive? Thankis a lot
    Last edited: Feb 28, 2010
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  3. Feb 28, 2010 #2


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    Yes. I don't know how to elaborate, but if you try to prove otherwise (integral = 0), you will get a contradiction.
  4. Feb 28, 2010 #3


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    Yes. Suppose f:A-->R is such a function. We have that


    and so by subadditivity of the measure

    [tex]\mbox{mes}(A)\leq \sum_n\mbox{mes}\left(f^{-1}\left(\left[\frac{1}{n},+\infty\right)\right)\right)[/tex]

    Since mes(A)>0, it must be that


    for some n. By definition


    where h:A-->R denotes a positive measurable stair function bounded above by f.

    It follows that

    [tex]\int_A f\geq \int_{A_n}\frac{1}{n}=\frac{1}{n}\mbox{mes}(A_n)>0[/tex]
  5. Feb 28, 2010 #4


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    i see. thanks a lot indeed.
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