If the weak limit is zero, is it true that the sequence of functions tends to zero almost everywhere?

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Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]tomathbbR$ has the following property. For any smooth function $phi:[0,1]tomathbbR$ one has
$$
lim_ntoinfty int_0^1phi(x) f_n(x),dxto0.
$$



Is it true that $f_n(x)to 0$ for almost all $xin[0,1]$.



Note that it is not assumed here that $f_n$ are bounded or positive




functional-analysis limits weak-convergence




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  • 1




    Look up the Riemann Lebesgue lemma.
    – copper.hat
    Aug 2 at 17:42










  • thanks! that was very helpful
    – Oleg
    Aug 2 at 17:46














up vote
1
down vote

favorite












Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]tomathbbR$ has the following property. For any smooth function $phi:[0,1]tomathbbR$ one has
$$
lim_ntoinfty int_0^1phi(x) f_n(x),dxto0.
$$



Is it true that $f_n(x)to 0$ for almost all $xin[0,1]$.



Note that it is not assumed here that $f_n$ are bounded or positive




functional-analysis limits weak-convergence




share|cite|improve this question















  • 1




    Look up the Riemann Lebesgue lemma.
    – copper.hat
    Aug 2 at 17:42










  • thanks! that was very helpful
    – Oleg
    Aug 2 at 17:46












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]tomathbbR$ has the following property. For any smooth function $phi:[0,1]tomathbbR$ one has
$$
lim_ntoinfty int_0^1phi(x) f_n(x),dxto0.
$$



Is it true that $f_n(x)to 0$ for almost all $xin[0,1]$.



Note that it is not assumed here that $f_n$ are bounded or positive




functional-analysis limits weak-convergence




share|cite|improve this question











Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]tomathbbR$ has the following property. For any smooth function $phi:[0,1]tomathbbR$ one has
$$
lim_ntoinfty int_0^1phi(x) f_n(x),dxto0.
$$



Is it true that $f_n(x)to 0$ for almost all $xin[0,1]$.



Note that it is not assumed here that $f_n$ are bounded or positive





functional-analysis limits weak-convergence





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 2 at 17:36









Oleg

302210




302210







  • 1




    Look up the Riemann Lebesgue lemma.
    – copper.hat
    Aug 2 at 17:42










  • thanks! that was very helpful
    – Oleg
    Aug 2 at 17:46












  • 1




    Look up the Riemann Lebesgue lemma.
    – copper.hat
    Aug 2 at 17:42










  • thanks! that was very helpful
    – Oleg
    Aug 2 at 17:46







1




1




Look up the Riemann Lebesgue lemma.
– copper.hat
Aug 2 at 17:42




Look up the Riemann Lebesgue lemma.
– copper.hat
Aug 2 at 17:42












thanks! that was very helpful
– Oleg
Aug 2 at 17:46




thanks! that was very helpful
– Oleg
Aug 2 at 17:46










1 Answer
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No. $f_n(x)=sin(nx)$ is a counterexample.






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    up vote
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    down vote



    accepted










    No. $f_n(x)=sin(nx)$ is a counterexample.






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      up vote
      2
      down vote



      accepted










      No. $f_n(x)=sin(nx)$ is a counterexample.






      share|cite|improve this answer























        up vote
        2
        down vote



        accepted







        up vote
        2
        down vote



        accepted






        No. $f_n(x)=sin(nx)$ is a counterexample.






        share|cite|improve this answer













        No. $f_n(x)=sin(nx)$ is a counterexample.







        share|cite|improve this answer













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        share|cite|improve this answer











        answered Aug 2 at 17:39









        David C. Ullrich

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