Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure).

It is well known that for a sequence $(f_n)$ in $L^1(\Omega)$ which converges to zero (in $L^1(\Omega)$, that is, $\|f_n\|_{L^1(\Omega)}\to 0$) then a subsequence of $f_n$ converges to zero pointwise almost everywhere. My question is: Is there a uniform version of this theorem in the following sense: For a countable set $A$ of sequences $(f_n)$ in $L^1(\Omega)$ and $$\sup_{(f_n)\in A} \|f_n\|_{L^1(\Omega)}\to 0$$ is it true that there exists a set of measure zero $N\subset \Omega$ and an increasing sequence of indices $(k_n)$ such that for $x\notin N$, $$\sup_{(f_n)\in A} |f_{k_n}(x)|\to 0$$ as $n\to \infty$?

Edit: "uniform convergence pointwise a.e." is obviously not such a great way to describe this kind of convergence, so if someone knows how this should be called, this might be of help too.