I was thinking about this fact:
Let $X$ be a measure space and let $f_n : X \rightarrow \mathbb{C}$ a sequence of measurable functions in $L^p$ which converges a.e. to a function $f$ in $L^p$. Is it true that $f_n$ converges to $f$ in the $L^p$ norm with $p<\infty$?
I did the case $p\geq 1$. It works, basically my idea was to use triangle inequality and Minkowski inequality to prove that $(|f_n|+|f|)^p$ is summable. From $|f_n-f|^p\leq (|f_n|+|f|)^p$ and the dominated convergence theorem (generalized version) we finish.
But what about the case $p<$1?