Suppose
(a) $f_n \to f$ a.e. on $[0,1]$ and
(b) $\|f_n\|_{L^\infty_{(0,1)}} \leq M \lt \infty.$
I want to show that $f$ is Lebesgue integrable and $f_n \to f$ in $L^1{[0,1]}$.
Also, I want an example where the condition (b) is not satisfied and the conclusion fails.