Let $p \in [1,\infty)$, $f,f_1,f_2,...\in \mathscr{L}^1(X,\mathscr{A},\mu, \mathbb{R})$ are all nonnegative functions, such that ${f_n}\rightarrow f$ a.e. It is known that this is not sufficient to guarantee that $\lim_n ||f_n-f||_p=0$. If I add the condition $\lim_n ||f_n||_p=||f||_p$, does it imply $\lim_n ||f_n-f||_p=0$ now? This seems intuitively appealing but I am not able to prove (or disprove) it.
Any suggestions?