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Math Help - convergence in L^2

  1. #1
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    convergence in L^2

    Let f_n \rightarrow f in L^2(X,F,m).
    If m(X)<\infty, then \int_X f_n \rightarrow \int_X f.
    any hint would be appreciated.
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  2. #2
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    Quote Originally Posted by Archi View Post
    Let f_n \rightarrow f in L^2(X,F,m).
    If m(X)<\infty, then \int_X f_n \rightarrow \int_X f.
    any hint would be appreciated.
    Cauchy–Schwarz (or Hölder): \Bigl|\int_X f_n - \int_X f\Bigr| \leqslant \int_X|( f_n-f)*1| \leqslant  \Bigl(\int_X |f_n-f|^2\Bigr)^{1/2}  \Bigl(\int_X 1^2\Bigr)^{1/2}.
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