Here you are another question in basic measure theory...

Let $f_k$ be a measurable sequence of functions on $(X,M,\mu)$ measure space. Suppose that $f_k$ *does not go to 0 a.e*.. Can I then find a set $A\subseteq X$ with positive measure and a subsequence $f_{k_j}$ and an $\varepsilon > 0$ such that $\liminf_j |f_{k_j}(x)| > \varepsilon$ foreach $x\in A$?