5-10

If p>0 and
\int_{E}|f-f_k|^p\to 0,
and
\int_{E}|f_k|^p\leq M 
for all k, show that
\int_{E}|f|^p\leq M.
Solution, Using the previous problem, we know that f_k\to^{m} f; in particular, there exist a subsequent f_{k_j}\to f a.e. in E. Then by Fatou's Lemma,
\int_{E}|f|^p=\int_{E}\lim\inf|f_{k_j}|^p\leq\lim\inf\int_{E}|f_{k_j}|^p\leq M.