For $f\in L^1(\RR^n)$, $\int_E\abs{f}$ is small when $E$ is the complement of a large ball and when $m(E)$ is small.Suggest a slogan
Let \(m\) be Lebesgue measure on \(\RR^n\), let \(f\in L^1(\RR^n)\), let \(\varepsilon> 0\), and for \(N\geq1\) let \(B_N\subseteq\RR^n\) be the open ball of radius \(N\) centered at \(0\).
Small tails. There exists \(N\) such that \(\int_{B_N^c} \abs{f} < \varepsilon\).
Absolute continuity. There exists \(\delta>0\) such that every Lebesgue measurable \(E\subseteq\RR^n\) with \(m(E) < \delta\) satisfies \(\int_E \abs{f} < \varepsilon\).
What do small tails and absolute continuity say about $f \in L^1$?