5-9

If p>0, and
\int_{E}|f-f_k|^p\to 0
as k\to\infty, show that f_k\to^{m}f on E (and thus there is a subsequence f_{k_j}\to f a.e. in E).
Solution: We prove by contrapositive. Suppose there is \epsilon>0 such that |X_k|\geq\epsilon for all k, where X_k=\{|f-f_k|>\epsilon\}. Then
\int_{E}|f-f_k|^p\geq\int_{X_k}|f-f_k|^p\geq\int_{X_k}\epsilon^p\geq\epsilon^{p+1}>0
for all k. This contradicts the assumption that\int_{E}|f-f_k|^p\to 0.