measure theory - Application of dominated convergence theorem

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.

Answer

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.