Limit of functions in $L^p$

Question

If $f_n \to f(x)$ a.e. and $\sup_n ||f_n||_p<\infty$, how to show $f$ is in $L^p$?

Can I argue that by:

$||f||_p\leq \liminf_n||f_n||_p\leq\sup_n ||f_n||_p<\infty$?

Answer 1

If $p=\infty$ is included into consideration, you should treat it as a separate case, in which no integrals are involved. Let $M=\sup \|f_n\|_\infty$ and argue that $|f|\le M$ a.e.

The case $1\le p<\infty$ is handled with the Fatou lemma, as a commenter noted. $$\int \lim |f_n|^p \le \liminf \int |f_n|^p \le \sup \int |f_n|^p $$