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Assume that a sequence of continuous functions $(f_n)$, where $f:[0,1]\to\mathbb{R}$ has the following property. For any smooth function $\phi:[0,1]\to\mathbb{R}$ one has $$ \lim_{n\to\infty } \int_0^1\phi(x) f_n(x)\,dx\to0. $$

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

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

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    $\begingroup$ Look up the Riemann Lebesgue lemma. $\endgroup$ – copper.hat Aug 2 '18 at 17:42
  • $\begingroup$ thanks! that was very helpful $\endgroup$ – Oleg Aug 2 '18 at 17:46
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No. $f_n(x)=\sin(nx)$ is a counterexample.

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