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Thread: Iterated Integral

  1. #1
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    Iterated Integral

    Dear readers,

    I want to write down a rigorous proof of the intuitive fact:

    $\displaystyle \int^{\infty}_{0}\int^{x}_{0} \frac{dF(t)}{dt}dtdx =\int^{\infty}_{0}xF(x)dx$

    Anyone knows how to formulate such a proof?

    Thnx
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  2. #2
    MHF Contributor

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    No, there is no such proof because the "intuitive fact" you give is not true.

    By the "Fundamental theorem of Calculus" $\displaystyle \int_0^x \frac{dF(t)}{dt}dtdx= F(x)- F(0)$.

    $\displaystyle \int_0^\infty\int_0^x \frac{dF}{dt}dtdx= \int_0^\infty (F(x)- F(0))dx$

    That does not exist unless F(0)= 0 in which case
    $\displaystyle \int_0^\infty\int_0^x \frac{dF}{dt}dtdx= \int_0^\infty F(x) dx$
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