I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise.
Suppose you have $(f_n)_{n\ \in \mathbb N}\subset L^1(\mathbb R^2)$ such that $f_n \to 0$ in $L^1(\mathbb R^2)$.
Is it true that there exists a subsequence $f_{n_k}$ such that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?