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

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