Exercise 23, § 5

Let (f_n) be a sequence of functions in $\mathscr{S}_F$ such that, for every integrable set A and every subsequence (f_{n_k}) of (f_n), there exists a subsequence of (f_{n_k}) that converges to 0 almost everywhere in A; show that the sequence (f_n) converges in measure to 0. (Argue by contradiction.)