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Thread: iterated integral

  1. #1
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    iterated integral

    Let $\displaystyle \lambda,\mu$ be Borel measures on $\displaystyle I=[0,1]$ for x-axis and y-axis. Let $\displaystyle d=\{(x,y): x<y\}\subset I\times I$.
    a)show $\displaystyle d$ is measurable on $\displaystyle I\times I$.
    b)Evaluate $\displaystyle \int_I \int_I 1_d d\lambda d\mu$, $\displaystyle \int_I \int_I 1_d d\mu d\lambda$, and $\displaystyle \int_{I\times I} 1_d d(\lambda \times \mu)$.

    i have no idea how to do this. any help would be appreciated.
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  2. #2
    Super Member girdav's Avatar
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    Let $\displaystyle f\left(x,y\right) =x-y$, for $\displaystyle x,y\in \mathbb R $ We have
    $\displaystyle d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I$ and because $\displaystyle f$ is measurable we have the result.
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  3. #3
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    Quote Originally Posted by girdav View Post
    Let $\displaystyle f\left(x,y\right) =x-y$, for $\displaystyle x,y\in \mathbb R $ We have
    $\displaystyle d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I$ and because $\displaystyle f$ is measurable we have the result.
    how do you show that $\displaystyle f$ is a measurable function? and i am still not sure how to do these iterated integrals. please help.
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  4. #4
    Super Member girdav's Avatar
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    It's the sum of two measurable functions.
    I think you can use Fubini because you have a $\displaystyle \sigma$-finite space $\displaystyle \left(I,\mu\right)$.
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