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Suppose $\{f_n\}$ is a sequence of functions in $L^2[0,1]$ such that $f_n(x)\rightarrow 0$ almost everywhere on $[0,1]$. If $\|f_n(x)\|_{L^2[0,1]}\le 1$ for all $n$ , then $$\lim_{n\rightarrow \infty} \int_0^1 f_n(x)g(x)~dx =0$$ for all $g\in L^2[0,1]$.

I feel I have to use dominated convergence theorem, but I can't fined the dominating functions. Thanks for helping.

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See here for ideas. –  David Mitra May 12 '12 at 2:01
    
My first instinct is to try applying Holder's inequality as $$\int_0^1 |f_n(x)g(x)|dx\leq \left(\int_0^1 |f_n(x)|^2dx\right)^{1/2}\left(\int_0^1 |g(x)|dx\right)^{1/2}$$ but I'm not sure if $\|f_n\|_2$ can be effectively bounded. –  Alex Becker May 12 '12 at 2:07
    
The problem is the sequence $f_n$ just converges almost everywhere and do not necessarily converges to zero in norm. –  matgaio May 12 '12 at 2:08
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2 Answers

up vote 2 down vote accepted

Here's an outline:

Fix $g\in L^2$.

Choose $\delta>0$ so that $\Bigl(\int_E|g|^2\Bigr)^{1/2}$ is small whenever $\mu(E)<\delta$.

By Egoroff, find a set $E$ of measure less than $\delta$ so that $f_n$ converges uniformly to $0$ off $E$. Choose $N$ so that for $n>N$, $|f_n|$ is small on $E^C$.

Then write: $$ \Bigl| \int f_n g\, \Bigr|\le \int |f_n ||g| =\int_E |f_n ||g|+\int_{E^C}|f_n ||g| $$ and apply Hölder to both integrals on the right.

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Try to apply Egorov's theorem.

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I'm not sure this works, because Egorov's leavues us with a set of $\epsilon>0$ measure left over and no bound on how $f_n$ behave on it. –  Alex Becker May 12 '12 at 2:18
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We can perhaps estimate $\|f_n\|_{L^2(E)}$ where $E$ is the small set where the uniform convergence fails. –  matgaio May 12 '12 at 2:22
    
@matgaio Ah, I think I see it. –  Alex Becker May 12 '12 at 2:28
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