equality of lim sup and lim

Question

If we know $f_n \rightarrow f$ almost everywhere and if we have :- $$\limsup_{n \to \infty} \int_X |f_n - f|^p \, d\mu \le 0 $$ How can we get $$\lim_{n \to \infty} \int_X |f_n - f|^p \, d\mu = 0$$

Remark:- the space $(X,M,\mu)$ is a measure space.

Answer 1

Seems to me that you do not need almost sure convergence of $f_n$ to $f$.

$$0\le\liminf_{n\rightarrow\infty}\int_X|f_n-f|^p\,\mathrm{d}\mu\le\limsup_{n\rightarrow\infty}\int_X|f_n-f|^p\,\mathrm{d}\mu\le0,$$

thus necessarily,

$$\liminf_{n\rightarrow\infty}\int_X|f_n-f|^p\,\mathrm{d}\mu=\limsup_{n\rightarrow\infty}\int_X|f_n-f|^p\,\mathrm{d}\mu=0.$$