We know that for $p\ge1$ , given a positive measure $\mu$ on a set $X$ if $f_n\to f$ almost everywhere, $f,f_n\in L^p(\mu)$ for all $n$, and $||f_n||_p\to ||f||_p$ as $n\to \infty$ then $||f-f_n||_p\to 0$ (See here)
Will this remain true for $0<p<1$?
The inequality we get here is $|f-f_n|^p<|f|^p+|f_n|^p$. I found this problem in big Rudin, where he suggests: By using Egorov's theorem write $X=A \cup B$ such that $\int_A |f|^p \le \varepsilon$, and $\mu(B)\le \infty$ and $f_n\to f$ uniformly on $B$ and then applying Fatou's lemma to $\int_B|f_n|^p$. This doesn't seem like a direct application of Egorov's since the space is not given to have finite measure.