Let $f_n : [0,1] \rightarrow \mathbf{R}$ be a sequence of measurable functions such that
$\bullet$ $f_n \rightarrow 0$ a.e. on $[0,1]$.
$\bullet$ $\int_{[0,1]} |f_n|^2 dm \leq 1$ for all $n \geq 0$.
Then I want to show that $\int_{[0,1]} |f_n| dm \rightarrow 0$ as $n \rightarrow \infty$.
I tried to combine Egorov's Theorem and Dominated Convergence Theorem but I could not find a dominating function for $|f_n|$.