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Thread: Norm of indicator function in L^2-space

  1. April 5th 2010, 09:48 AM #1
    Swlabr
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    Norm of indicator function in L^2-space

    I am reading a book by one John Conway, and at one point he seems to imply that \|\chi_{\Delta}\|_2 = \left(\int |\chi_{\Delta}|^2 d \mu\right)^{1/2} = (\mu({\Delta}))^{1/2} (where \chi_{\Delta} is the indicator function).

    I can't quite understand this. I feel it should just be \mu({\Delta})...

    So, does anyone know why?

    Thanks in advance!
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  2. April 5th 2010, 10:18 AM #2
    karkusha
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    Because {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta, and \int \chi_\Delta d\mu=\mu(\Delta)
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  3. April 5th 2010, 10:28 AM #3
    Swlabr
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    Quote Originally Posted by karkusha View Post
    Because {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta
    Yes, of course! Thanks!
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  4. April 5th 2010, 10:32 AM #4
    Swlabr
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    Quote Originally Posted by karkusha View Post
    Because {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta, and \int \chi_\Delta d\mu=\mu(\Delta)
    Wait a tick - the square is outside the absolute value, not inside...
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  5. April 5th 2010, 10:38 AM #5
    karkusha
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    |\chi_\Delta|^2=|{\chi_\Delta}^2|=|\chi_\Delta|=\c hi_\Delta
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