real analysis - A convergence result for functions in L^2

Let $(f_n)_{n\geq 1}$, $f$ and $g$ be functions in $L^2$ [0,1].
Suppose $f_n \rightarrow f$ pointwise almost everywhere.

If $|f_n(x)| \leq |x|^{-1/3}$ prove that :

$\lim_{n \to \infty} \int_0^1 f_n(x)g(x) = \int_0^1 f(x)g(x)$.

To me this looks very much like monotone convergence, but the existence of $g$ and the fact that the sequence may not be monotonic causes problems for me

Any help would be greatly appreciated.