Generality about dominated convergence theorem, does $|f_n|\leq g\in L^p$ and $f_n(x)\to f(x)$ a.e. tel us that $f_n\to f$ in $L^p$

Question

I have a big doubt about something, let $f_n\in L^p$ s.t. $|f_n|\leq g\in L^p$ and $f_n(x)\to f(x)$ a.e. Do we have that $f_n\to f$ in $L^p$ ? The only thing I get is $$\lim_{n\to \infty }\int |f_n|^p=\int |f|^p,$$ but I can't do better. May-be my result is wrong ?

Answer 1

Yes, it's true but your proof is not correct. You have that $(f_n-f)^p\in L^1$ and converge a.e. to $0$. Now, $$|f-f_n|^p\leq (|f_n|+|f|)^p \leq 2^p|g|^p\in L^1,$$ and thus, using dominated convergence theorem, you get $$\lim_{n\to \infty }\int (f_n-f)^p=0,$$ what prove the claim.