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Math Help - L2 space

  1. #1
    GTO
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    L2 space

    Let f_n,f,g \in L^2(X,F,\mu) and |f_n|\leq g for all n.
    Show that f_n\rightarrow f in L^2 iff f_n \rightarrow f in measure and \int f_n^2d\mu\rightarrow \int f^2d\mu.
    i could show that if f_n\rightarrow f in L^2 , \int f_n^2d\mu\rightarrow \int f^2d\mu.
    but im stuck on showing if f_n\rightarrow f in L^2, f_n \rightarrow f in measure. please help me on this. thank you.
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  2. #2
    GTO
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    Quote Originally Posted by GTO View Post
    Let f_n,f,g \in L^2(X,F,\mu) and |f_n|\leq g for all n.
    Show that f_n\rightarrow f in L^2 iff f_n \rightarrow f in measure and \int f_n^2d\mu\rightarrow \int f^2d\mu.
    i could show that if f_n\rightarrow f in L^2 , \int f_n^2d\mu\rightarrow \int f^2d\mu.
    but im stuck on showing if f_n\rightarrow f in L^2, f_n \rightarrow f in measure. please help me on this. thank you.
    Since f_n \rightarrow f in L^2, we have \int_X |f-f_n|^2 d\mu \rightarrow 0. Let E=\{x:|f-f_n|>\epsilon\} and D= \{x:|f-f_n|<\epsilon\}. Then \int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0. So \int_E |f-f_n|^2 d\mu\rightarrow 0.

    i got this far but i dont know how i can show |f-f_n|<\infty. i am thinking that the given assumption |f_n|<g can be used here but i dont know how i can use it. please help me .
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  3. #3
    GTO
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    Quote Originally Posted by GTO View Post
    Since f_n \rightarrow f in L^2, we have \int_X |f-f_n|^2 d\mu \rightarrow 0. Let E=\{x:|f-f_n|>\epsilon\} and D= \{x:|f-f_n|<\epsilon\}. Then \int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0. So \int_E |f-f_n|^2 d\mu\rightarrow 0.

    i got this far but i dont know how i can show |f-f_n|<\infty. i am thinking that the given assumption |f_n|<g can be used here but i dont know how i can use it. please help me .
    Can i assume that f_n is bounded since \int |f_n|^2 <\infty?
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  4. #4
    GTO
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    Quote Originally Posted by GTO View Post
    Since f_n \rightarrow f in L^2, we have \int_X |f-f_n|^2 d\mu \rightarrow 0. Let E=\{x:|f-f_n|>\epsilon\} and D= \{x:|f-f_n|<\epsilon\}. Then \int_X |f-f_n|^2 d\mu=\int_E |f-f_n|^2 d\mu +\int_D |f-f_2|^2 d\mu \rightarrow 0. So \int_E |f-f_n|^2 d\mu\rightarrow 0.

    i got this far but i dont know how i can show |f-f_n|<\infty. i am thinking that the given assumption |f_n|<g can be used here but i dont know how i can use it. please help me .
    i am not really sure but since \int |f-f_n|^2 < \epsilon for all n>N, |f-f_n|<\infty. otherwise this integral will not go to 0. can someone tell me if it is correct?
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