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Math Help - iterated integral

  1. #1
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    iterated integral

    Let \lambda,\mu be Borel measures on I=[0,1] for x-axis and y-axis. Let d=\{(x,y): x<y\}\subset I\times I.
    a)show d is measurable on I\times I.
    b)Evaluate \int_I \int_I 1_d d\lambda d\mu, \int_I \int_I 1_d d\mu d\lambda, and \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 f\left(x,y\right) =x-y, for x,y\in \mathbb R We have
    d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I and because 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 f\left(x,y\right) =x-y, for x,y\in \mathbb R We have
    d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I and because f is measurable we have the result.
    how do you show that 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 \sigma-finite space \left(I,\mu\right).
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