I want to prove that given a sequence [MATH]f_n [/MATH] in [MATH]L^p[/MATH]. If [MATH]f_n \rightarrow f [/MATH] a.e. and [MATH]\sup_n \| f_n\| _ p < \infty[/MATH], then [MATH]f\in L^p[/MATH] and [MATH]\|f\|_p \le \liminf_{n\rightarrow \infty} \| f_n \|_p [/MATH].
I'm really lost. I'm stuck on this exercise since yesterday, but the only thing I managed to understand is that I think I should use Fatou's lemma to prove the inequality.
I'm really lost. I'm stuck on this exercise since yesterday, but the only thing I managed to understand is that I think I should use Fatou's lemma to prove the inequality.