It is well known that tail of an integrable function on $\mathbb{R}^d$ is small, i.e., Given $\epsilon>0$, there is $R>0$ such that $$\int_{\{|x|>R\}}|f(x)|dx<\epsilon.$$
I was wondering whether there were any simple conditions under which a family or a sequence of integrable functions $\{f_n\}$ would have small tails. Something like: given $\epsilon>0$, there is $R>0$ such that $$\int_{\{|x|>R\}}|f_n(x)|dx<\epsilon~\mbox{for all}~n.$$
It looks similar to uniform integrability but there we have integration on a finite set.
The property seems to behave bad with uniform integrability (take $f_n:=n\chi_{B(0,1)}$).
If $S_N:=\{ x,N\leqslant x<N+1\}$, then your condition means that $$\sup_n\sum_{k\geqslant L}a(n,k)\to 0,$$ where $a(n,k):= \int_{S_k}|f_n|$. We can't say that $\{(a(n,k),k\geqslant 1),n\geqslant 1\}$ is compact in $\ell^1$ because boundedness is missing.