I have the following question about $L^p$ spaces:
Suppose that $f,f_1,f_2,\ldots$ are functions in $L^p$ for some $p \geq 1$ and the sequence converges in $L^p$ to $f$, i.e. $||f_n-f||_p \to 0$.
Does this imply that the sequence $|f_1|^p,|f_2|^p,\ldots$ converges in $L^1$ to $|f|^p$?
Is it also true if we remove the absolute value? That is, the sequence $f_1^p,f_2^p,...$ converges to $f^p$, in case $p$ doesn't create problem with power (for example $p=3$).
I am using necessary and sufficient condition from the Vitali's convergence theorem and it seems to work.
Do you have some advices or some references for this result?